Symmetry and Condensed Matter Physics Unlike existing texts, this book blends for the first time three topics in physics: symmetry, condensed matter physics, and computational methods, into one pedagogical textbook. It includes new concepts in mathematical crystallography; experimental methods capitalizing on symmetry aspects; nonconventional applications such as Fourier crystallography, color groups, quasi-crystals and incommensurate systems; and concepts and techniques behind the Landau theory of phase transitions. The textbook adopts and develops a computational approach to the application of group theoretical techniques to solving symmetry-related problems. This dramatically alleviates the need for intensive calculations, even for the simplest systems, usually found in the presentation of symmetry. Writing computer programs helps the student achieve a firm understanding of the underlying concepts, and sample programs, based on Mathematica , are presented throughout the book. Containing over 150 exercises, this textbook is ideal for graduate students in condensed matter physics, materials science, and chemistry. Solutions and computer programs are available online at www.cambridge.org/9780521828451. Mi c h a e l El-Bata n o u n y is a Professor in the Department of Physics at Boston University. His research area is experimental surface physics and he has written numerous papers on solid state physics and surface physics. Fr e d e r i c k Wo o t e n (1928–2004) was Professor of Physics and Chair of the Department of Applied Science at the University of California, Davis. He is the author of Optical Properties of Solids and numerous articles in the fields of solid state physics and materials science.
Symmetry and Condensed Matter Physics A Computational Approach M. El-Batanouny Boston University
F. Wooten University of California, Davis
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521828451 © M. El-Batanouny and F. Wooten 2008 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008
ISBN-13 978-0-511-39482-9
eBook (NetLibrary)
ISBN-13
hardback
978-0-521-82845-1
Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
This book is dedicated to my loving wife Gloria, to the memory of my parents Maurice and Hayat, and to my sister Nagwa. (ME)
Contents
Preface
page xi
1
Symmetry and physics 1.1 Introduction 1.2 Hamiltonians, eigenfunctions, and eigenvalues 1.3 Symmetry operators and operator algebra 1.4 Point-symmetry operations 1.5 Applications to quantum mechanics Exercises
1 1 5 9 14 17 19
2
Symmetry and group theory 2.1 Groups and their realizations 2.2 The symmetric group 2.3 Computational aspects 2.4 Classes 2.5 Homomorphism, isomorphism, and automorphism 2.6 Direct- or outer-product groups Exercises
21 21 25 30 34 41 42 43
3
Group representations: concepts 3.1 Representations and realizations 3.2 Generation of representations on a set of basis functions Exercises
51 52 62 67
4
Group representations: formalism and methodology 4.1 Matrix representations 4.2 Character of a matrix representation 4.3 Burnside’s method Exercises Computational projects
69 69 78 85 94 95
5
Dixon’s method for computing group characters 5.1 The eigenvalue equation modulo p 5.2 Dixon’s method for irreducible characters
96 96 99
vii
viii
Contents 5.3 Computer codes for Dixon’s method Appendix 1 Finding eigenvalues and eigenvectors Exercises Appendix 2 Computation project
107 116 128 128 133
6
Group action and symmetry projection operators 6.1 Group action 6.2 Symmetry projection operators 6.3 The regular projection matrices: the simple characteristic Exercises
134 134 138 157 162
7
Construction of the irreducible representations 7.1 Eigenvectors of the regular Rep 7.2 The symmetry structure of the regular Rep eigenvectors 7.3 Symmetry projection on regular Rep eigenvectors 7.4 Computer construction of Irreps with dα >1 7.5 Summary of the method Exercise
164 164 168 170 172 176 178
8
Product groups and product representations 8.1 Introduction 8.2 Subgroups and cosets 8.3 Direct outer-product groups 8.4 Semidirect product groups 8.5 Direct inner-product groups and their representations 8.6 Product representations and the Clebsch–Gordan series 8.7 Computer codes 8.8 Summary Exercises
179 179 179 185 190 191 192 208 214 215
9
Induced representations 9.1 Introduction 9.2 Subduced Reps and compatibility relations 9.3 Induction of group Reps from the Irreps of its subgroups 9.4 Irreps induced from invariant subgroups 9.5 Examples of Irrep induction using the method of little-groups Appendix Frobenius reciprocity theorem and other useful theorems Exercises
217 217 217 219 230 247 257 261
Crystallographic symmetry and space-groups 10.1 Euclidean space 10.2 Crystallography 10.3 The perfect crystal 10.4 Space-group operations: the Seitz operators 10.5 Symmorphic and nonsymmorphic space-groups 10.6 Site-symmetries and the Wyckoff notation
263 263 272 273 294 297 323
10
Contents
ix
10.7 Fourier space crystallography Exercises
347 357
11
Space-groups: Irreps 11.1 Irreps of the translation group 11.2 Induction of Irreps of space-groups Exercises
362 362 377 407
12
Time-reversal symmetry: color groups and the Onsager relations 12.1 Introduction 12.2 The time-reversal operator in quantum mechanics 12.3 Spin-1/2 and double-groups 12.4 Magnetic and color groups 12.5 The time-reversed representation: theory of corepresentations 12.6 Theory of crystal fields 12.7 Onsager reciprocity theorem (Onsager relations) and transport properties Exercises
409 409 409 419 429 442 458
13
Tensors and tensor fields 13.1 Tensors and their space-time symmetries 13.2 Construction of symmetry-adapted tensors 13.3 Description and classification of matter tensors 13.4 Tensor field representations Exercises
474 474 487 498 536 551
14
Electronic properties of solids 14.1 Introduction 14.2 The one-electron approximations and self-consistent-field theories 14.3 Methods and techniques for band structure calculations 14.4 Electronic structure of magnetically ordered systems Appendix 1 Derivation of the Hartree–Fock equations Appendix 2 Holstein–Primakoff (HP) operators Exercises
552 552 552 566 618 632 633 636
15
Dynamical properties of molecules, solids, and surfaces 15.1 Introduction 15.2 Dynamical properties of molecules 15.3 Dynamical properties of solids 15.4 Dynamical properties of surfaces Appendix 1 Coulomb interactions and the method of Ewald summations Appendix 2 Electronic effects on phonons in insulators and semiconductors Exercises
638 638 638 668 702 704
Experimental measurements and selection rules 16.1 Introduction 16.2 Selection rules 16.3 Differential scattering cross-sections in the Born approximation
716 716 717 721
16
464 472
708 713
Contents
x
18
16.4 Light scattering spectroscopies 16.5 Photoemission and dipole selection rules 16.6 Neutron and atom scattering spectroscopies Exercises 17.1 Phase transitions and their classification 17.2 Landau theory of phase transitions: principles 17.3 Construction and minimization techniques for ∆Φ Exercises
727 739 748 775 777 781 830 856
Incommensurate systems and quasi-crystals 18.1 Introduction 18.2 The concept of higher-dimensional spaces: superspaces and superlattices 18.3 Quasi-crystal symmetry: the notion of indistinguishability and the classification of space-groups 18.4 Two-dimensional lattices, cyclotomic integers, and axial stacking
858 858
Bibliography References Index
865 879 887 901 903 912
Preface
Pedagogical presentation and analysis of the symmetry aspects of physical systems in terms of group theoretical concepts and methodology has been evolving over the past six or seven decades, since the pioneering textbooks by Weyl and Wigner first appeared. This constantly evolving pedagogy has resulted in over a hundred textbooks on the subject. The impetus behind these efforts has stemmed from the general recognition of the invaluable role that the application of such methodology plays in determining and predicting the properties of a physical system. Symmetry concepts provide a very useful means for systematizing the description of a physical system in terms of its energy and momentum, and other relevant physical quantities. Furthermore, the incipient methodologies furnish a very efficient framework for classifying its physical states, and a crucial machinery for simplifying the intervening numerical applications of physical laws. By means of the irreducible representations of its symmetry group, one can classify physical states and particles in a logical way and establish selection rules, which predict restrictions on possible transitions between different physical states. The use of symmetry also simplifies numerical calculations, for example, in solving the Schr¨ odinger equation for condensed matter systems. Moreover, from the symmetry properties of a physical system, one can make conclusions about the values of measurable physical quantities, and, conversely, one can trace a symmetry group of a system from observed regularities in measured quantities. There is also an intimate connection between symmetry, invariance and dynamical laws. Despite all these merits, the application of symmetry methods has been seriously hampered by the need for painstaking and tedious calculations, even when dealing with the simplest group symmetries. This drawback has led many students and practitioners in the area of condensed matter physics to shy away from learning the fundamentals of this invaluable discipline. In the recent past, a number of new ideas, techniques and approaches have emerged that have yet to be incorporated in textbooks on symmetry. We believe that the adoption of these novel aspects within a new pedagogical approach would dramatically simplify existing instructional procedures, remove the cumbersome calculational barriers that tend to severely limit the scope of examples and exercises, and provide a conduit for elucidating and unifying seemingly diverse aspects under one umbrella. We attempt to implement such an approach in this book, and present the necessary tools for its application in diverse areas in condensed matter physics. xi
xii
Preface
One of the novel features characterizing this book and setting it apart from others is that it adopts and develops at the outset a computational approach to the theory of finite groups and its applications in physics. A computational approach to group theory has several significant advantages. First, it eliminates traditional difficulties encountered in problem solving. Second, it provides an alternative pedagogical process whereby a student would learn the material through writing computer programs: A logical prerequisite for program writing is that the student must have acquired a detailed understanding of the material at hand! The modern basis for computational methods in group and representation theories dates from the 1960s, with the development of computer algorithms for generating permutation groups and studying their structure, and with the introduction of John Dixon’s method for the exact calculation of “characters” of finite groups, which can be easily implemented on a computer. Currently, there are at least two computational systems that are well suited for the analysis of symmetry groups: GAP and MAGMA; they can be accessed through the web. However, here we shall follow the methods proposed by Stig Flodmark and Esko Blokker. These authors, recognizing the importance and utility of Dixon’s method, developed computer algorithms and programs to implement Dixon’s exact character method, and to use the results to construct the corresponding unitary irreducible matrix representations. We develop the underlying ideas, algorithms and methodology for such calculations in the first seven chapters. The reader is first exposed to the relevant conceptual aspects, then introduced to corresponding computational algorithms, and instructed in methods for implementing these algorithms into programs and subroutines. We find that Mathematica, because of its capability to handle symbolic and combinatorial manipulations, provides a natural and convenient environment for the development of such programs. So, the main instructions in the computational approach will be based on the language of Mathematica. However, every student will be encouraged to develop programs in any language he or she is comfortable with. Our versions of all the Mathematica-based computational programs will be posted on a website, together with data files relevant to space- and point-groups. Our work is intended to make the computational approach to group theory available to a wider audience. It is aimed at students in the physical sciences: physics, chemistry, materials science and, possibly, some disciplines in engineering. We also have in mind the working professional who would like to learn the subject or who already knows it but is unfamiliar with the modern computational approach. In addition to adopting computational techniques, we introduce and develop several concepts that have, at best, been marginally treated in textbooks on symmetry, to date. We develop the ideas of group actions on systems and their decomposition into orbits and strata. We demonstrate and stress the fundamental relevance of the study of the corresponding orbit space and of the set of strata to physical problems. For example, we demonstrate how the notions of the star of the wavevector, which appears in the theory of representations of space-groups, of Wyckoff positions, which are encountered in crystallography, and of images of the order-parameter of a phase transition are all manifestations of orbits. Thus, we inadvertently apply these concepts to different domains of condensed matter physics, without realizing that they are actually decompositions into orbits and strata made under different names, and, in fact they should be unified under
Preface
xiii
one umbrella. Thus, linear representations, which are engendered by group action on a basis set, are just a particular case of group actions. Group actions are introduced in Chapter 6. Subsequently, their use and appearance are stressed when appropriate. In Chapter 6, we also introduce the concept of symmetry-projection operators and develop the computational tools for their construction, using their hermiticity and idempotency properties to cast the corresponding matrices into the form of a simple eigenvalue problem. In Chapter 8, we introduce the notions of subgroups, cosets, normal subgroups, product groups, Kronecker products, and Clebsch–Gordan series; and end by presenting techniques for determining the Clebsch–Gordan coefficients. In Chapter 9, we explore the processes of induction and subduction of irreducible representations (Irreps) and the concept of compatibilities relating these Irreps. In Chapter 10, we present a long and comprehensive expos´e of crystallography. The novel feature of this chapter is the detailed presentation of concepts in mathematical crystallography. These include a detailed discussion of arithmetic holohedries and classes, as well as Bravais classes, classification of space-groups with respect to affine conjugations, site-symmetries, and Wyckoff positions and sets. Another unique feature of this chapter is a section on Fourier crystallography, which is one of the two methods used to study the symmetries of quasi-crystals in Chapter 18. In Chapter 11, we develop the machinery for determining the Irreps of symmorphic and nonsymmorphic space-groups. In Chapter 12, we introduce time-reversal symmetry, and discuss the concepts of doublegroups. We include in this chapter a long section on color, or Shubnikov, groups and another section detailing the construction of corepresentations. The section on the color groups includes dichromatic (black and white) point-groups, lattices and space-groups. It also discusses the extension of these concepts to polychromatic symmetries. Another section on crystal-field theory is presented and extended to cover the case of dichromatic symmetry. We end this chapter with a detailed discussion of the manifestation of time-reversal symmetry in transport properties, which is elegantly cast in the Onsager reciprocity relations. The remaining chapters, 13–18, are dedicated to applications in diverse fields of condensed matter physics. We have extended the applications beyond the usual topics taken up in most texts on the subject. In Chapter 13 we develop the theory of tensors, present techniques for the construction of symmetry-adapted tensors, and finally present a catalog of the different material tensors. We end this chapter with an expos´e of tensor fields and their relation to symmetry projection operators. Chapter 14 develops the basic principles of the electronic structure of solids and methods for computation. The final section presents how the special features of the electronic structure of magnetically ordered systems can be viewed and classified in terms of corepresentations. Chapter 15 develops methods for computing and classifying the dynamical properties of solids and solid surfaces. Chapter 16 is dedicated to the discussion of symmetry-based experimental techniques such as neutron and atom scattering, angle-resolved photoemission, and Raman spectroscopy.
xiv
Preface
The Landau theory of phase transitions is discussed in detail in Chapter 17. Applications to commensurate–incommensurate and magnetic phase transitions are given. A long section is devoted to methods of construction of the Landau free energy, including ones based on the theory of invariant polynomials, group action, and order-parameter image groups. Finally, Chapter 18 addresses the fundamental aspects of the symmetry of quasi-crystals and incommensurate systems. The symmetries of these systems are developed along two tracks. First, the concepts of hyperspace symmetries and their projection onto the natural lower-dimensional spaces are presented. Second, the application of the techniques of Fourier space crystallography, developed in Chapter 10, are discussed. The main emphasis, throughout this book, is on exposing the conceptual building blocks of this mathematical theory of symmetry. Consequently, we frequently skip over long detailed mathematical proofs which can be found in a multitude of textbooks. Instead, we point the reader to references where such proofs are clearly presented. Moreover, in contrast to previous texts on the subject, the different aspects of group theory are presented wherever they are needed rather than being lumped into one single expos´e. This has the merit of associating a certain aspect of the theory with a tangible physical attribute. Thus, for example, we defer the introduction of cosets and invariant subgroups till we discuss space-groups, where they find immediate application. Our interest throughout is in clarity and simplicity rather than elegance. We have striven to meet the needs of the beginner who must work through the gory details of many simple examples, much as we ourselves did in trying to learn the subject matter. We do not wish to hide these details in compact mathematical notation.
1 Symmetry and physics
1.1 Introduction The application of group theory to study physical problems and their solutions provides a formal method for exploiting the simplifications made possible by the presence of symmetry. Often the symmetry that is readily apparent is the symmetry of the system/object of interest, such as the three-fold axial symmetry of an NH3 molecule. The symmetry exploited in actual analysis is the symmetry of the Hamiltonian. When alluding to symmetry we usually include geometrical, time-reversal symmetry, and symmetry associated with the exchange of identical particles. Conservation laws of physics are rooted in the symmetries of the underlying space and time. The most common physical laws we are familiar with are actually manifestations of some universal symmetries. For example, the homogeneity and isotropy of space lead to the conservation of linear and angular momentum, respectively, while the homogeneity of time leads to the conservation of energy. Such laws have come to be known as universal conservation laws. As we will delineate in a later chapter, the relation between these classical symmetries and corresponding conserved quantities is beautifully cast in a theorem due to Emmy Noether. At the day-to-day working level of the physicist dealing with quantum mechanics, the application of symmetry restrictions leads to familiar results, such as selection rules and characteristic transformations of eigenfunctions when acted upon by symmetry operations that leave the Hamiltonian of the system invariant. In a similar manner, we expect that when a physical system/object is endowed with special symmetries, these symmetries forge conservation relations that ultimately determine its physical properties. Traditionally, the derivation of the physical states of a system has been performed without invoking the symmetry properties, however, the advantage of taking account of symmetry aspects is that it results in great simplification of the underlying analysis, and it provides powerful insight into the nature and the physics of the system. The mathematical framework that translates these symmetries into suitable mathematical relations is found in the theory of groups and group representations. This is the subject we will try to elucidate throughout the chapters of this book. Let us begin with a tour de force, exploring the merits of invoking symmetry aspects pertinent to familiar but simple problems. We start by reminding ourselves of the trivial example of using symmetry, or asymmetry, to simplify the evaluation of an integral. 1
Symmetry and physics
2 Consider
+b
sin x dx = 0. −b
We know this to be true because sin x is an odd function; sin(−x) = − sin(x). In evaluating this integral, we have taken advantage of the asymmetry of its integrand. In order to cast this problem in the language of symmetry we introduce two mathematical operations: I, which we will identify later with the operation of inversion, and which, for now, changes the sign of the argument of a function, i.e. I f (x) = f (−x); and E, which is an identity operation, E f (x) = f (x). This allows us to write +b b b sin x dx = (E + I) sin(x) dx = 1 + (−1) sin(x) dx = 0. −b
0
0
Figure 1.1 shows schematically the plane of integration, with ⊕ and indicating the sign of the function sin x. We may introduce a more complicated integrand function f (x, y), and carry the integration over the equilateral triangular area shown in Figure 1.2.
y
–b b
x
Fig. 1.1. The asymmetric function sin x. y
x
Fig. 1.2. Integration domain.
1.1 Introduction
3
Making use of the 3-fold symmetry of the triangle, which includes rotations by multiples of 2π/3, as well as reflections shown in Figure 1.3, we write the integral in the form f (x, y) dx dy = E + O(2π/3) + O(4π/3) + σ1 + σ2 + σ3 triangle
wedge
× f (x, y) dx dy, where the Os represent counterclockwise rotations by the angle specified in the suffix, and the σs are defined in Figure 1.3. Now, if the function f (x, y) possesses a symmetry which can be associated with that of the triangle, as for example shown in Figure 1.4, the integral vanishes. Later, we will see how to reach similar conclusions in the case of selection rules, for example, where the situation may be much more complicated. Next, we present a simple example to demonstrate how to invoke symmetry to simplify the solution of dynamical problems. We consider a system of two masses and three springs as illustrated in Figure 1.5. Assume both masses to be equal to m and that all springs have the force constant κ. In that case, the Hamiltonian, which is the
y 4p/3
2p/3 x
y s3
s2 x s1
Fig. 1.3. Symmetry operations of an equilateral triangle.
+
–
v2 v
–
+ +
1 –
Fig. 1.4. Some possible symmetries of f (x, y) on an equilateral triangle.
Symmetry and physics
4
x1
x2
Fig. 1.5. A two-mass system with inversion symmetry about the center point. total energy of the system, H=
2 2 p2i κ κ + x2i + x2 − x1 , 2m 2 2 i=1
is invariant under the operation of inversion. That is, an inversion of the system through the mid-plane, which takes x1 → −x2 and x2 → −x1 , leaves the Hamiltonian invariant. A lengthy normal mode analysis shows that the energy eigenvectors are 1 1 and . 1 −1 This means there are two distinct modes of vibration, one in which the masses move in opposite directions and by equal amounts and one in which the masses move in the same direction and by equal amounts. We can exploit the symmetry of this problem to obtain the same result, but with much less effort. Let I be the inversion operator. Since the Hamiltonian is invariant under the inversion operation, I commutes with H and thus the eigenvectors of I are also the eigenvectors of H. Writing the displacements x1 and x2 as the components of a vector u we have x1 −x2 Iu = I = . (1.1) x2 −x1 If the vector u is an eigenvector of I then we have −x2 Iu = = λ u, −x1
(1.2)
where λ is the eigenvalue, and applying the inversion operation once more we obtain I 2 u = λ2 u = I(Iu) = I
−x2 −x1
=
x1 x2
= u.
(1.3)
Thus λ2 = 1, and the eigenvalues of I are λ = ±1, with corresponding eigenvectors 1 1 u1 = and u2 = , (1.4) 1 −1 which are identical to those of the Hamiltonian. They describe the displacement amplitudes of the masses during normal mode vibrations and are either even or odd with respect to the inversion symmetry.
1.2 Hamiltonians, eigenfunctions, and eigenvalues
5
We learn that for a classical system with a symmetric one-dimensional potential V (x), the eigenfunctions are of either odd or even parity, and we have seen how this knowledge can be used to simplify finding the solution to a problem. Quantum systems are generally much more complicated. Except for a few important but relatively simple systems, such as the hydrogen atom, the harmonic oscillator, and the Kronig–Penney model for a periodic lattice, most problems in quantum mechanics must be solved numerically. Because the calculations are often long and tedious, and much effort is devoted to numerical methods, accuracy becomes a concern. Fortunately, some simplifications based on symmetry can be rigorously made. These usually involve the construction of symmetry projection operators, which are, in turn, based on the concept of irreducible representations (Irreps) and their characters. We will develop the ideas of characters and representations in Chapters 3–5. Here we state only that they play an all-important role in group theory. We will present in Chapters 5 and 7 exact computational procedures for the calculation of matrix Irreps and characters using a method proposed by John Dixon in 1967. However, we stress here only that all the important quantitative symmetry information can be obtained with the aid of simple computer calculations.
1.2 Hamiltonians, eigenfunctions, and eigenvalues A typical problem in condensed matter physics involves the determination of the physical states of a system given its Hamiltonian or its free energy. Consequently, the symmetry we need to exploit is, generally, that of the Hamiltonian. The application of symmetry will then require the definition of the corresponding operators and the specification of the rules of the action of these operations. Since a Hamiltonian can be defined on the configuration space of the physical system, we define a symmetry operation as a transformation effected on this configuration space. An example is the inversion operator I, which effects a vector transformation r → −r. Other typical operations are reflections, rotations, and translations. We write the transˆ as formation involving a symmetry operator O ˆ H O. ˆ O We start by collecting the symmetry operations that leave the Hamiltonian H invariant, i.e. those whose action leaves H unchanged. We designate this set of operations the ˆ is such an operator, then symmetry group1 of H. If O ˆHO ˆ† = H O
(1.5)
ˆ H = H O. ˆ O
(1.6)
or ˆ leaves the Hamiltonian invariant, it must commute with the Hamiltonian. But Thus, if O what is the effect on an eigenfunction belonging to H? We have H Ψi = Ei Ψi 1
The word group will be justified in Chapter 2.
(1.7)
Symmetry and physics
6
y
y
Cˆ4 +
– x
+
x
– py
px
Fig. 1.6. Cˆ4 acting on py produces px . ˆ gives which, when operated on by O, ˆ (H Ψi ) = O ˆ (Ei Ψi ). O
(1.8)
ˆ commutes with H, as described by (1.6), we obtain Since the symmetry operator O ˆ Ψi ), ˆ Ψi ) = Ei (O H (O
(1.9)
ˆ i is also an eigenfunction of the same and we see that if Ψi is an eigenfunction, OΨ Hamiltonian with the same eigenvalue Ei . From quantum mechanics it is known that two commuting operators share a common set of eigenfunctions. Here we learn that by operating on an eigenfunction with a symmetry operator we generate another eigenfunction belonging to the same eigenvalue. A pictorial example of the generation of a new eigenfunction with a symmetry operator is shown in Figure 1.6, it depicts the generation of a px -orbital by operating on a py -orbital with a π/4 rotation about the z-axis. This example illustrates that the symmetry operators also act on functions defined on configuration space.2 ˆ and all other All functions generated by the successive application of the operator O symmetry operators of the system form a degenerate manifold in the Hilbert space of the Hamiltonian. The family of functions forming the basis of this manifold is classified by some characteristic symmetry properties, which distinguish it from other classes or families of functions in the Hilbert space. To illustrate this property of generating distinct families of functions in Hilbert space, we consider the consequence of space isotropy, which is encountered under many guises 2
A note on function-space operators Function-space operators are generally denoted by a “hat” over the symbol for the operator, ˆ This is not formally necessary in that it is possible to infer the nature of the operator as in O. by its context. If an operator acts on a function it is a function operator. Thus, sometimes, to reduce clutter, it may be convenient to not use a hat. Indeed, we do not use a hat over the Hamiltonian, which operates on wavefunctions. However, prudent practice is to use the hat for emphasis, since there are subtle differences in the action of a function operator on the coordinate variables of a function and the action of a configuration-space operator on the coordinates of a physical entity. This will be made clear in later sections of this chapter.
1.2 Hamiltonians, eigenfunctions, and eigenvalues
7
in mechanics, electromagnetics, and quantum mechanics. It is manifest in the family of functions we know as spherical harmonics, Ylm , which form disjoint subspaces denoted by the index l, each subspace l has dimension 2l + 1.3 Since isotropy means invariance under arbitrary infinitesimal rotations, it implies that a system with such symmetry should be invariant under all possible rotation operations. In quantum mechanics, isotropic symmetry of a system implies that all such operations should commute with its Hamiltonian, and thus their application to the eigenfunctions of the Hamiltonian should reveal the degeneracy within each spherical harmonic manifold in its Hilbert space. In classical mechanics and electromagnetics, as well as in quantum mechanics, the closure of the different manifolds is demonstrated by the well-known addition theorem of Legendre polynomials. In essence these relations express the fact that pure angular rotations of, for example, p-states about their origin result in another p-state and thus do not alter the angular characteristics of the eigenfunction. Similar rotations of a dipole produce only other dipoles, not other multipoles. The existence of a set of classes of symmetries to which symmetrized functions distinctly belong in isotropic space, that is, the spherical harmonics, is not an isolated case. It is the general feature for any underlying configurational symmetry. For example, the homogeneity of space leads to the natural adaptation of plane-wave functions as a distinct set for classifying symmetrized functions in this configuration space. The symmetry classes associated with a given symmetry group are conventionally known as its irreducible representations. Another example may be found in quantum systems for which the symmetry of indistinguishability among a set of particles leads to two distinct and disjoint classes of wavefunctions which are commonly referred to as fermion and boson wavefunctions, or particle systems. To underscore the connection between the symmetry and the nature of the allowed states of such systems let us review the argument behind this well-established classification. The indistinguishability of the particles requires that the associated Hamiltonian should not change when two particles are exchanged, that is, H is invariant under such an exchange, or permutation, operation. If we denote the operator associated with a twoparticle permutation operation by Pˆ12 , we may write Pˆ12 H Ψ = H Pˆ12 Ψ = E Pˆ12 Ψ.
(1.10)
Now, the commutation of Pˆ12 with H requires that they share common eigenfunctions. If we consider the eigenfunctions of Pˆ12 , we have Pˆ12 Ψ = λ Ψ,
(1.11)
which means the exchange operation leaves the state unchanged apart from a multiplicative, possibly complex, factor λ. A second application of Pˆ12 returns the system to its original configuration, so that 2 Pˆ12 Ψ = λ2 Ψ = Ψ, 3
(1.12)
In electromagnetics the spherical harmonics describe monopole, dipole, quadrupole, . . . fields, and in atomic physics they are manifest as s, p, d, . . . states. Indeed, the structure of the periodic table of the elements is a consequence of the three-dimensional isotropic character of space.
8
Symmetry and physics
hence λ2 = 1
(1.13)
or λ = ±1, thus establishing two distinct classes of eigenstates of a Hamiltonian describing a set of indistinguishable particles, namely, those that transform even and those that transform odd under the permutation or particle-exchange operation.
1.2.1 Examples of symmetry and conservation laws Translation and conservation of linear momentum Given a classical system’s Hamiltonian H(x, p), where x and p are conjugate coordinate and momentum variables, we write the canonical equations of motion as ∂H , ∂p ∂H . p˙ = − ∂x
x˙ =
If the system is invariant under arbitrary translations, then the r.h.s of the second equation vanishes and the linear momentum p is conserved. When the system is treated quantum mechanically, we define the translation operator corresponding to an infinitesimal displacement dx as R(dx) x = x + dx, and, as we will show in Section 1.3, its action on function-space is given by
ˆ ψ(x) = ψ(x − dx) = ψ(x) −i p dx ψ(x) , R ˆ can be expressed as so that the operator R p dx ˆ R = exp −i . If the Hamiltonian is invariant under such operations, then
ˆ H = 0. R,
(1.14)
(1.15)
Using Heisenberg’s equation of motion and substituting (1.14), we obtain
1 d p = p, H = 0. dt i
(1.16)
Inversion and parity conservation Consider a system which remains invariant under the inversion operation, i.e.
ˆ H = 0, I, then we find that it also commutes with the time-translation operator
H (t2 − t1 ) ˆ ˆU ˆ = 0. U (t2 , t1 ) = exp −i ⇒ I,
(1.17)
1.3 Symmetry operators and operator algebra
9
Since Iˆ commutes with H, they have simultaneous eigenfunctions, such that
Iˆ ψ(t1 ) = λ1 ψ(t1 ) .
(1.18)
Since Iˆ2 is the identity, λ1 assumes the values ±1 only, and we obtain
ˆ (t2 , t1 ) ψ(t1 ) = U ˆ (t2 , t1 )Iˆ ψ(t1 ) Iˆ ψ(t2 ) = Iˆ U
ˆ (t2 , t1 ) ψ(t1 ) = λ1 ψ(t2 ) . = λ1 U
(1.19)
1.3 Symmetry operators and operator algebra We introduced earlier the idea that symmetry operators may act on Hamiltonians as well as functions, and that these actions can be regarded as transformations in configuration space. Here, we shall give a detailed expos´e of these ideas, and establish some concepts and conventions needed in order to be self-consistent in applying symmetry operations to physical applications of group theory.
1.3.1 Configuration-space operators Configuration space is the real physical space in which physical objects move, i.e. where we define classical particle trajectories and quantum mechanical wavefunctions and probabilities. Operators acting in this space are known as configuration-space operators. Now, we consider a point object P, a particle in this space, located at position r with respect to the reference axes, as shown in Figure 1.7. If the object is translated by a vector t, it will arrive at the new position defined by the vector r with respect to the axes such that r = r + t.
(1.20)
We can thus define an active operator tA , and its inverse t−1 A such that tA r = r + t t−1 A r
= r − t.
(1.21)
y
y
tA
P r x
t
r r
P x
Fig. 1.7. Active operator tA translates the point body P from r to r .
Symmetry and physics
10
y
y
y
tP
P r
P r
x
t
r
x, x
Fig. 1.8. Passive operator tP translates the coordinate system by −t.
We show a grid of equivalent points in configuration space because we have in mind, say, a crystal structure in which all the grid points correspond to atomic positions. In such a case, moving one point (atom) P of a body (a crystal) moves all the points. The space is infinite, as is the (idealized) crystal. In Figure 1.7, we show the identical fixed coordinate system twice in order to see the before and after pictures. Now consider instead the effect of the passive operator, tP , which translates the coordinate axes by −t while keeping the point object fixed, as in Figure 1.8. Then the new position of the point P after the translation is still given by tP r = r = r + t,
(1.22)
but the interpretation is different. The new position is with reference to the new x , y coordinate system. The operator tP has the inverse effect from that of tA . Likewise, rotation operations can be either passive or active, bearing inverse relations to each other. 1.3.2 Function-space operations We have alluded to a connection between configuration-space operators and function-space operators. As a prelude to showing that a coordinate transformation in configuration space induces a transformation of the function defined on it, we consider the case of a traveling wave. This contains the basic physics and mathematics in a familiar and easily understood situation. Consider a transverse wave of shape f (x, t) traveling with positive velocity along the x-axis, as shown in Figure 1.9. If we are to deal with the wave motion quantitatively, we need a mathematical description of the shape of the wave moving as a whole. We assume no damping or dispersion, so the shape remains unchanged, and any point on the wave may be taken as a reference point. We will follow the motion of the wave peak. To describe an amplitude that remains constant, although its position moves to the right with increasing time, we need a function of the general form f (x, t) = f (x − vt).
(1.23)
With a function of this form, as t increases, x also increases, but the increase in x is exactly compensated for in the argument of f by the term −vt, so that the argument is
1.3 Symmetry operators and operator algebra
11
y x = vt
x x
Fig. 1.9. Profile of a traveling wave at (x, t) = (0, 0) and at (x , t ). y
y
y
tˆP x
t
x, x
Fig. 1.10. Function operator tˆP generates a new function.
unchanged: There is an inverse relation between transformations in configuration space (the wave moving to the right with velocity v) and transformations in function-space (the argument of the function being translated to the left by −vt). Now consider a function that describes the contour of an atomic orbital. We take that function to be a circle centered on the atom at the origin of the coordinate system in Figure 1.10 and given by the equation x2 + y 2 = c2
(1.24)
f (x) = y = (c2 − x2 )1/2 .
(1.25)
or
The transformation tP translates the x, y coordinate system such that the point with coordinates (x, y) in that system has coordinates (x , y ) in the translated coordinate system, as given by x = x + t, y = y.
(1.26)
Along with this transformation of coordinates is a change in the description of the function (the atomic orbital) associated with the lattice point. It is now described as f (x ) = y = (c2 − (x − t)2 )1/2 ,
(1.27)
Symmetry and physics
12 y
y
tˆA x
t
x
Fig. 1.11. tˆA generates a new function.
as can easily be verified. Here, we are viewing the same function from two different coordinate systems. We have f (x) as the function in the x, y coordinate system and f (x ) in the x , y coordinate system, but the physical point (the atom) has not moved, only a coordinate shift has been made. We describe it as a new function in Figure 1.10 only in the sense that it has a new functional form in the new coordinate system. There is an alternative viewpoint, and it is the one we will use. That is to recognize f (x ) as a new function and treat it as being generated by the action of a function operator tˆA acting on the original function f (x), as illustrated in Figure 1.11. From this viewpoint we want to know the function operator that generates the new function f (x ) induced by the transformation in configuration space. Just as in the treatment of a wave traveling to the right, as in Figure 1.9, we write the new function in terms of the one fixed coordinate system, but include a coordinate shift to keep the value of the new function, centered at the new coordinates, equal to the value of the original function centered on the original coordinates. Thus we have 2 2 1/2 tˆA f (x) = f (t−1 . A x) = (c − (x − t) )
(1.28)
In this active description, tˆA f is a new function that is defined in terms of (1.23). That is, we cannot define the function operator directly, but only in terms of its action on the original function, taking account of the inverse action on the coordinates in the argument of the function by means of the operation t−1 A x. These are important concepts, to which we will return in Chapter 3, after first becoming familiar with the basic elements of group theory and symmetry operations in Chapter 2.
Example 1.1 From (1.25), the value of the function f (x) at x = 0 is f (0) = ±c.
1.3 Symmetry operators and operator algebra
13
The new function, tˆA f (x), evaluated at the new coordinate x = t, is, from (1.28), tˆA f (x) = (c2 − (x − t)2 )1/2 = (c2 − (t − t)2 )1/2 = ±c, illustrating that the new function tˆA f (x) evaluated at the new transformed coordinates equals the original function f (x) evaluated at the original coordinates.
1.3.3 Operator algebra Here we wish to consider two important but not always immediately obvious properties of operator algebra. Suppose we were to operate on the product of two functions f (x) ∗ g(x). How do we ˆ (x) ∗ g(x))? The meaning of R ˆ is that it operates on the whole carry out the operation R(f space on which the functions f (x) and g(x) are defined. Thus the whole space for both functions must be transformed, giving ˆ (x) ∗ g(x)) = Rf ˆ (x) ∗ Rg(x). ˆ R(f
(1.29)
Another tricky bit of algebra is the application of one function-space operator after another. It must be done with great care. The general definition of the product of operators of any type requires that, in the present case, ˆ Sf ˆ (x) = R( ˆ Sf ˆ (x)), R
(1.30)
ˆ is applied on the function Sf ˆ of x, the functional symbol being which means that R ˆ is itself a new function, obtained by operenclosed in brackets for emphasis. That is, Sf ˆ We could write this new function as ating on f (x) with S. ˆ (x) = g(x). Sf
(1.31)
Then ˆ Sˆ f (x) = R ˆ g(x) = g(R−1 x) R = Sˆ f (R−1 x) = f (S −1 R−1 x) = f ((RS)−1 x).
(1.32)
Thus if RS = T , then ˆ Sˆ f (x) = f ((RS)−1 x) = f (T −1 x) = Tˆ f (x), R
(1.33)
ˆ Sˆ = Tˆ, RS = T → R
(1.34)
whence
which is the very important isomorphism of the multiplication rules of function-space and configuration-space operators.
14
Symmetry and physics 1.4 Point-symmetry operations
Point-symmetry operations are performed about a fixed point in configuration space. They include rotations about an axis passing through the point, reflections through planes containing the point, and inversions through the point. These operations are unitary, i.e. ˆO ˆ† = O ˆ†O ˆ = ˆl (the identity operator in matrix form). O In the following we describe the various point-symmetry operations and introduce their commonly used notations. The Schoenflies notation is given first, followed by the International (or Hermann–Mauguin) notation in parentheses. Our interest here is in point-groups, for which the Schoenflies notation is usually used. However, the Hermann–Mauguin symbols can also describe the far more complicated symmetry of crystals and space-groups. We will revisit Hermann–Mauguin notation in Chapter 10. (i) Identity E, for einheit, (1) This operation leaves an object fixed or rotated by 2π about some axis. (ii) Proper rotations Cnk (n) An object is rotated 2kπ/n clockwise about an axis that passes through the fixed point. The axis of highest order n, if unique, is designated as the principal axis. It is usually taken as the z-axis. The remaining axes are usually labeled by primes or by x or y if they coincide with either the x- or y-axes. In three-dimensional Euclidean space, the rotation of a vector r by 2kπ/n about the z-axis is defined by the rotation matrix ⎤ ⎡ cos(2kπ/n) sin(2kπ/n) 0 ⎥ ⎢ Cnk ≡ R(2kπ/n) = ⎣− sin(2kπ/n) cos(2kπ/n) 0⎦ . (1.35) 0 0 1 These matrices have a determinant of +1 ⎡ cos(2kπ/n) ⎢ Cnk Cnk† = ⎣− sin(2kπ/n) 0 ⎡ cos(2kπ/n) ⎢ × ⎣ sin(2kπ/n) 0
and satisfy the unitarity property ⎤ sin(2kπ/n) 0 ⎥ cos(2kπ/n) 0⎦ 0 1 ⎤ − sin(2kπ/n) 0 ⎥ (1.36) cos(2kπ/n) 0⎦ = E. 0 1
The discreteness of molecular and crystallographic systems limits n to integer values in these cases. For molecules, in principle, there is no restriction on n. However, a crystal lattice restricts the values of n to 1, 2, 3, 4, and 6. The symbols used to denote the allowed n-fold rotation axes in two- and threedimensional space lattices are shown in Figure 1.12. Note the absence of five-fold symmetry, as well as symmetries higher than six-fold. The restriction can be verified as follows. If a lattice is composed of equilateral m-sided polygons, the angle subtended by two sides at a vertex is π (m − 2) , with m ≥ 3, m
1.4 Point-symmetry operations
C2
C3
C4
15
C6
Fig. 1.12. Symbols for rotation axes.
Cn
Cn sv
sh sd (a)
(b)
Fig. 1.13. Reflection operations.
since m < 3 is not a physically meaningful polygon. Because a symmetry rotation about the vertex must leave the lattice unchanged, we have 2π (m − 2) π = , n m
or
n=
2m , m−2
(1.37)
which yields, for integer n, m = 3, 4, 6, n = 6, 4, 3.
(1.38)
A lattice of (nonequilateral) rectangles provides n = 2. In order to change the orientation of a rotation axis one has to employ a unitary operator transformation Cn = RCn R† .
(1.39)
(iii) Reflection operations Reflection operations are illustrated in Figure 1.13. Part (a) shows reflection through a plane perpendicular to a principal axis σh (1/m). When the rotation axis is along the z-direction, σh is defined by the matrix ⎤ ⎡ 1 0 0 ⎥ ⎢ (1.40) σh = ⎣0 1 0 ⎦. 0 0 −1
Symmetry and physics
16
Part (b) shows reflection through a plane containing a principal axis σv (m). If the plane bisects the angle between two 2-fold axes perpendicular to a principal axis, the reflection is denoted by σd (m). (iv) Inversion I (¯ 1) Inversion takes a spatial vector r into −r. In three-dimensions it is defined by the matrix ⎤ ⎡ −1 0 0 ⎥ ⎢ (1.41) I = ⎣ 0 −1 0⎦ . 0 0 −1
Example 1.2 In order to change the orientation of a Cn rotation axis along the z-axis to one along the x-axis we choose R to be a counterclockwise rotation of π/2 about the y-axis. Thus, ⎤ ⎤ ⎡ ⎤⎡ ⎡ 0 0 1 cos(2π/n) sin(2π/n) 0 0 0 −1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ Cnx = ⎣0 1 0 ⎦ ⎣− sin(2π/n) cos(2π/n) 0⎦ ⎣ 0 1 0⎦ −1 0 0 0 0 1 1 0 0 ⎤ ⎡ 1 0 0 ⎥ ⎢ = ⎣0 cos(2π/n) − sin(2π/n)⎦. 0 sin(2π/n) cos(2π/n)
(v) Improper rotations Sn (¯ n) Improper rotations consist of proper rotations Cn followed by reflections σh , with symbols as shown in Figure 1.14. An improper rotation is defined by Sn = Cn σh = σh Cn .
(1.42)
Since σh2 = E, we find that Snk = Cnk ,
S2
S3
for even k.
S4
S6
Fig. 1.14. Improper rotations.
(1.43)
1.5 Applications to quantum mechanics
17
Another interesting property of improper rotation-reflection operations is that ⎡
−1 ⎢ S2 = C2 σh = ⎣ 0 0
0 −1 0
⎤⎡ 1 0 ⎥⎢ 0⎦ ⎣0 0 1
0 1 0
⎤ ⎡ −1 0 ⎥ ⎢ 0 ⎦=⎣ 0 0 −1
0 −1 0
⎤ 0 ⎥ 0 ⎦ = I, −1
(1.44)
or C2 I = σh , which we use to obtain S3 = C3 C2 I = C65 I = IC6−1 , S4 = C4 C2 I = IC43 = IC4−1 , S6 = C6 C2 I = IC64 = IC3−1 .
1.5 Applications to quantum mechanics We now consider, briefly, two applications of group theory to quantum mechanics, in order to give a bit more insight into the physical applications of group theory before immersing ourselves in the details of group theory itself. The first application concerns selection rules; the second is the simplification of the search for eigenfunctions in complicated systems. Selection rules determine which transitions between eigenstates are allowed by the application of a given perturbation operator to the system. They are concerned with the vanishing or nonvanishing of matrix elements (integrals) associated with operators representing external perturbations. The integrals may be evaluated analytically or numerically, whichever is more feasible, in order to determine whether a transition is allowed or not. However, we may notice that these integrals have the general form F (r) dr = F (R−1 r) d(R−1 r) = F (R−1 r) dr, (1.45) where the last step in (1.45) follows because the orthogonal matrices that describe transformations between orthogonal coordinate systems are such that the absolute value of the determinant of the orthogonal transformation matrix must be unity, that is, |det R| = 1. This means that lengths and angles are preserved in the transformation so that, in this case, the volume element d(R−1 r) equals the volume element dr.
Example 1.3 A one-dimensional configuration space may have reflection symmetry about a given point. F may be even or odd with respect to the reflection operation associated with this symmetry, F + (−x) = F (x), F − (−x) = −F (x), thus, as we know from freshman calculus, ∞ F − (x) dx = 0; −∞
∞ −∞
F + (x) dx = 0.
(1.46)
(1.47)
Symmetry and physics
18
Another way to arrive at this simple result is to invoke the reflection symmetry operation σ ˆ to the integrand F , such that σ ˆ F (x) = F (−x). Taking the symmetry point as the origin we obtain ∞ ∞ F (x) dx = [F (x) + σ ˆ F (x)] dx, −∞
(1.48)
0
which leads to the same result as using (1.47). We designate F + to be invariant under the operation of reflection in one dimension.
Thus we find, once again, that if we consider a one-dimensional oscillator, its eigenstates can be classified into two groups under the reflection operation: odd or even. The oscillator may be subjected to an external perturbation, κx, which has a dipolar or vector character, in that it is an odd function under the reflection operator. The transition probabilities are then evaluated through the integral ∞ ψf∗ (x) (κx) ψi (x) dx, (1.49) −∞
where ψi and ψf are the initial and final states, respectively. This integral can be cast in the form of (1.48), namely, ∞ ∞ ψf∗ (x) (κx) ψi (x) dx = [ψf∗ (x) (κx) ψi (x) + σ ˆ (ψf∗ (x) (κx) ψi (x))] dx −∞
0
∞
=
[ψf∗ (x) (κx) ψi (x) + ψf∗ (σ −1 x) (κσ −1 x) ψi (σ −1 x)] dx,
(1.50)
0
which vanishes when both ψi and ψf are either odd or even. The second example we consider is the simplification of the search for eigenvalues. ˆH = HR ˆ written in a matrix repWe start by considering the commutation relation R ˆ where R ˆ is a symmetry operator much like the resentation based on eigenfunctions of R, operator tˆA that we considered earlier. Then, ˆ ij Hjk = ˆ jk Hij R (1.51) R j
j
with ˆ |j = cj |j. R
(1.52)
ˆ Imposing a requirement of orthonormality for the set of eigenstates |j of R, ˆ ij = i|R|j ˆ = i|cj |j = cj δij . R
(1.53)
Thus, the summation in (1.51) reduces to a single term, yielding ˆ ii Hik = Hik R ˆ kk , R which can be rewritten as
ˆ kk ˆ ii − R R
(1.54)
Hik = 0.
(1.55)
Exercises
19
ˆ This simple result is of great significance. If i and k refer to different eigenvalues of R, then ci = ck , and Hik = 0. This means we will often be able to limit our considerations to different classes of functions separately. That is, because there are no off-diagonal elements of H connecting functions of different symmetry (Hik = 0), we will be able, for example, to limit the search for eigenfunctions that diagonalize the Hamiltonian operator by making ˆ our search separately within the classes of functions having different eigenvalues of R. Group theory is particularly helpful in simplifying large matrices by converting the large matrix into one of block diagonal form, with the elements in each of the smaller blocks belonging to eigenfunctions of the same symmetry. Thus, for example, we will find 3 × 3 and 5 × 5 blocks associated with p-orbitals and d-orbitals, respectively. Our goal is to be able to utilize group theory in the computational solution of physical problems. The intellectual pleasure that derives from the intrinsic elegance of the theory is a bonus. The practical reward that comes from learning group theory from a computational viewpoint is the ability to solve large problems on a computer, problems whose solution would not be feasible using the traditional tools of paper and pencil. Thus, after an introduction to abstract group theory and the realization of groups and some traditional examples involving the manipulation of 3 × 3 matrices, we will learn to do group theory calculations on a computer – to do them exactly, unhindered by the size of the matrices involved.
Exercises 1.1 Write down and solve the equations of motion for the system of masses and springs shown in Figure 1.5. Assume both masses to be equal and all springs to have the same force constant. Show that the eigenvalues for the energy are given by ω 2 m = k, 3k, from which the eigenvectors can be found to be in agreement with the results obtained purely by symmetry arguments. Must all three force constants be equal for this result to be obtained? Can you decide based on symmetry arguments alone? 1.2 Write the equations of motion for Exercise 1.1 in the form M u = −ω 2 mu, where M is a matrix. Use the two eigenvectors found in the text, 1 1 and u2 = , u1 = 1 −1, to construct the matrix
1 S= 1
1 , −1
where the first column of the matrix is given by u1 and the second column, by u2 . Find S −1 and then diagonalize M according to S −1 M S = λI, where I is the unit matrix, to find the eigenvalues λ. 1.3 Find the function generated by C4 acting on the function xf (r). n/2 1.4 Show that the operation Sn = I for n/2 odd. Show that for even n, Sn implies the existence of Cn/2 .
Symmetry and physics
20
z
C2
y x
Fig. 1.15. A C2 rotation about an axis bisecting opposite edges of a tetrahedron. C3 z
C3[111]
C3 y x
Fig. 1.16. A C3 rotation about a [111] body-diagonal axis of a cube.
1.5 Consider the case of an n-fold principal axis. Show that the introduction of a two-fold symmetry axis perpendicular to it implies the coexistence of n equivalent axes for n odd, and the coexistence of n/2 equivalent axes for n even. 1.6 Introduce diagonal reflection planes, σd , in the previous problem and show, using the three-dimensional defining matrices for a σd plane and a neighboring two-fold axis C2 , that the product C2 σd = S2n . 1.7 Show that the determinant of an improper rotation is −1. 1.8 Obtain the three-dimensional rotation matrix for the C2 axis joining two opposite edges of a tetrahedron, shown in Figure 1.15. Note that the origin of the coordinate system is the centroid of the tetrahedron. (Hint: Start with a two-fold rotation about the z-axis, then use a counterclockwise rotation about the x-axis to transform the axis to its final position, as shown in the figure.) 1.9 Using the results of the previous problem, find the new function generated by the ˆ2 acting on zf (r). function operator C 1.10 Obtain the three-dimensional rotation matrix for the operation C3 [111] shown in Figure 1.16. (Hint: Start with a three-fold rotation axis along the z-direction, followed by rotating the axis counterclockwise, by the same angle as in Exercise 1.8, but this time about the y-axis. Finally, rotate the axis counterclockwise by 45◦ about the z-axis.)
2 Symmetry and group theory
2.1 Groups and their realizations Abstract group theory defines relationships among a set of abstract elements in terms of binary operations among the elements of the group. The operations are known as group multiplication. Formally, a group consists of a set of elements with the following properties: (i) The product of any two elements in the set is a member of the set. Thus the set is closed under all group multiplication operations. (ii) If A, B, and C are elements of the group, then A(BC) = (AB)C. The associative law of multiplication holds; the commutative law of multiplication need not hold. (iii) There is a unit element, an identity element, E such that EA = AE = A. (iv) Each element A has a unique inverse A−1 such that AA−1 = A−1 A = E. A typical abstract group multiplication table is given in Table 2.1, for the group we denote by G6 , which consists of six elements. The convention for such tables is that the ij th element in the table is the product of the ith element labeling the rows and the j th element labeling the columns. From the table we see that AB = D, which means that the operation B followed by the operation A is equivalent to the single operation D. Note that AB = D, but BA = F ; thus AB = BA in this case. Abelian groups If XY = Y X for all elements of the group, the group is called Abelian. It is clear from the asymmetry about the diagonal of Table 2.1 that the group G6 is not Abelian. Table 2.1. Multiplication table for the abstract group G6 .
E A B C D F
E E A B C D F
A A E F D C B
B B D E F A C 21
C C F D E B A
D D B C A F E
F F C A B E D
22
Symmetry and group theory
Group order The number of elements in a group is called the order of the group and is designated by g. When g is finite the group is designated a finite group. Here g = 6. Cyclic groups There is one other abstract group of order 6: E, A, A2 , A3 , A4 , A5 , A6 . Groups comprising of E, A, . . . , Ai , . . . , An−1 are cyclic groups of order n. Group rearrangement theorem Every column and every row of the multiplication table contains each element of the group once and only once. Multiplication of every element in a row or column by some element of the group other than the identity element simply rearranges the elements. This is known as the group rearrangement theorem. If an element of a finite group is repeatedly multiplied by itself, it will eventually yield the identity. Thus, for example, from Table 2.1 we see that D3 = D(D2 ) = DF = E. The order of an element X is the smallest positive integer n such that X n = E. The order of D is 3. A physical example of an abstract group is called a realization. Distinct matrices representing distinct geometric transformations whose operations satisfy the group multiplication table are examples of realizations. Groups of numbers could also constitute a realization, but we will consider a realization of an abstract group to mean the elements of the group have a physical meaning. In order to realize a group, one has to: (i) Choose a relevant object or set of objects. (ii) Choose a characteristic of these objects. Coordinates, spin, or color are possibilities. (iii) Assign a physical meaning to the elements (the operations). Rotations, reflections, translations, and permutations are typical examples. The NH3 molecule is an instructive example of a physical object whose symmetry operations have a physical meaning and provide a realization of the group G6 . The three H atoms form an equilateral triangle. The N atom lies on the symmetry axis that passes through the center of the triangle and which is perpendicular to the plane of the triangle. The symmetry operations consist of the identity, rotations of ±120◦ about the symmetry axis passing through the N atom, and reflections through planes defined by the symmetry axis and one of the H atoms. Thus the symmetry operations of the NH3 molecule are the covering operations of the equilateral triangle for the case in which the triangle is restricted to remain in its original plane. By operating on the objects and observing the resulting changes, one can induce a possible realization of the group if there is a one-to-one correspondence between distinct operations and distinct outcomes. We induce a realization of the abstract group G6 using the covering operations of the NH3 molecule, Figure 2.1. These are the set of symmetry operations that send a figure into itself. The presence of mirror symmetry planes and a symmetry axis in the NH3 molecule must be reflected in the electronic structure and vibrational properties of NH3 , for these properties must be described by a Hamiltonian that is invariant under every symmetry coordinate transformation that leaves the molecule unchanged. Thus the covering operations of the equilateral triangle provide one of the simplest nontrivial examples of the connection between group theory and the symmetry properties of physical systems.
2.1 Groups and their realizations
23
Fig. 2.1. The NH3 molecule. s1
s3
A
C s2
B
Fig. 2.2. The symmetry planes (axes) of an equilateral triangle. The axes are considered fixed in space, not attached to the triangle. The symbol C3v denotes the point-group to which both the NH3 molecule and the equilateral triangle belong. The subscript 3 denotes a three-fold axis of symmetry perpendicular to the plane of the triangle and the subscript v denotes reflection-symmetry planes through the axis of symmetry and perpendicular to the plane of the triangle. To examine the point-group C3v , we must first identify and enumerate all possible symmetry operations that leave the triangle unchanged, or invariant. These operations may be identified with the help of Figure 2.2. They may be classified as (i) E, the identity element, which leaves the triangle unchanged, (ii) Two three-fold rotations: C3 ≡ (2π/3) and C32 ≡ (4π/3), are clockwise rotations about an axis normal to the plane of the triangle and passing through its center, in a conventional (right-hand screw) coordinate system. (iii) Three reflections through planes formed by the rotation axis and the perpendicular bisectors of the triangle: σ1 , σ2 , σ3 . We now define group multiplication of two elements as the outcome of the sequential application of the two operators. We find that both C3 ≡ (2π/3) and C32 ≡ (4π/3) are mutual inverses under this multiplication law, since when they are applied sequentially they are equivalent to E. Thus, the set of operations C3 ≡ (2π/3), C32 ≡ (4π/3) and E satisfy all the group specifications listed above. They satisfy the right-hand group multiplication table in Table 2.2. Similarly, we pair the identity element E with each of the σi s, to form three groups of order 2, each satisfying the left-hand table in Table 2.2. Moreover, we find the group
Symmetry and group theory
24
Table 2.2. Multiplication table for the groups C3 and Cs . E E C3 C32
E C3 C32
C3
C32
C3 C32 E
C32 E C3
E σi
E
σi
E σi
σi E
Table 2.3. Multiplication table for the group C3v .
E σ1 σ2 σ3 C3 C32
E
σ1
σ2
σ3
C3
C32
E σ1 σ2 σ3 C3 C32
σ1 E C32 C3 σ3 σ2
σ2 C3 E C32 σ1 σ3
σ3 C32 C3 E σ2 σ1
C3 σ2 σ3 σ1 C32 E
C32 σ3 σ1 σ2 E C3
criteria are fulfilled by the entire set of six elements. Thus, the group of six symmetry elements for the equilateral triangle is composed of the union of four smaller groups known as subgroups. The binary multiplications of the symmetry operations of the equilateral triangle, defined by the successive application of pairs of these operations to the triangle, are given in Table 2.3 for the group C3v . We can now make a one-to-one correspondence between the symmetry elements of C3v and the abstract group elements of Table 2.1 as follows: (i) The notation for the identity E is the same for both G6 and C3v . (ii) The operations A, B, and C are in one-to-one correspondence with the set of reflections through the symmetry planes shown in Figure 2.2 and denoted by (σ1 , σ2 , σ3 ). (iii) The operations D and F are in one-to-one correspondence with rotations of the triangle clockwise through 120◦ and 240◦ , respectively. The standard notation is C3 and C32 . If we compare Tables 2.1 and 2.3, we find they are identical, they differ only in the labeling of the group elements, but Table 2.3 represents a physical realization of the group. The one-to-one correspondence between elements in the two tables defines an isomorphic relationship. In C3v we have found a tangible realization of the abstract group G6 . One can label the vertices of a cardboard cutout of an equilateral triangle to experiment with the operations of the group, although with a bit of imagination the actual cutout is not needed. It is easy to verify the multiplication table. However, the group C3v is not the only realization of G6 . Another realization can be (1) (2) (3) obtained by replacing the reflections σ by two-fold rotations, C2 , C2 , and C2 , about
2.2 The symmetric group
25
the three perpendicular bisectors of the equilateral triangle. This is the dihedral group D3 associated with a two-faced equilateral triangle.
2.2 The symmetric group Still another realization of this group is defined by a set of permutations among three objects, as illustrated in Figure 2.3 and expressed in (2.1). These operations satisfy the same multiplication table as G6 and C3v when we define multiplication as the successive application of the permutations of the three objects. We may choose the three objects to be the three vertices of a triangle. Then, for example, we may identify the operation that takes vertex 1 into vertex 2, vertex 2 into vertex 3, and vertex 3 into vertex 1 to be equivalent to C3 of C3v , as shown in Figure 2.3. This group realization is called the symmetric group of order 3 and is denoted by S3 . There are 3! such permutations and we can identify their correspondences with the operations of C3v as follows: 1 2 3 1 2 3 , E ⇔ , σ1 ⇔ 1 3 2 1 2 3 1 2 3 1 2 3 , σ3 ⇔ , (2.1) σ2 ⇔ 3 2 1 2 1 3 1 2
C3 ⇔
2 3
3 , 1
C32
⇔
1 3
2 1
3 . 2
The one-to-one correspondence between the two sets of operations belonging to the symmetric group S3 and the point-group C3v defines an isomorphic mapping between them. The set of permutation operations obeys the group multiplication table given in Table 2.3 for the group C3v . One can think of objects (the vertices of the triangle in this case) as being allocated to boxes; with exactly one object per box. Then a permutation is just a rearrangement of the objects among the boxes. The notation of the permutation operations, given in (2.1), is such that we view the top row as labels of the three objects according to their initial locations in three boxes, thus, box 1 contains object 1, etc. The bottom row then denotes
2
1 2
C3
2
3
3
1
Fig. 2.3. The configuration space operator C32 rotates a triangle 120◦ clockwise.
Symmetry and group theory
26
the box destination of these objects as defined by the permutation operation. Then a permutation matrix describes the operation of taking the object in the top box of a column to the position (the box) just below it. We define the product of two permutations, say σ1 σ2 , as the permutation given by σ2 followed by the permutation given by σ1 .
Example 2.1 From Table 2.3 we see that σ1 σ2 = C3 . If we tions of (2.1), we get 1 2 3 1 2 1 3 2 3 2
represent these operations by the permuta 3 1 = 1 2
3 . 1
2 3
A very simple procedure can be used to obtain such a permutation: 1 2 3 1 2 3 3 2 1 1 2 3 ⇒ ⇒ ; 1 3 2 3 2 1 1 3 2 2 3 1 we replaced the top row of the left matrix by the lower row of the right matrix, then rearranged the resulting intermediate matrix such that the top row is again cast in ascending order.
After experimenting with such operations and checking with the multiplication table to ensure agreement, one soon recognizes the pattern for writing down the final result in standard form. Let us demonstrate that the associative law holds for these permutation operations. We ask, “Is σ1 (σ2 C3 ) = (σ1 σ2 )C3 ?” Representing these operations by permutations, we have 1 2 3 1 2 3 1 2 3 1 3 2 3 2 1 2 3 1 1 1
2 3
3 1 2 2
2 1
3 1 = 3 3
1 1
2 3
3 1 2 3
2 2
3 1 1 2
2 3
3 1 1 2
2 3
3 1 = 1 3
= Similarly,
=
1 2
2 1
3 . 2 3 1
2 3 2 1
3 . 2
Cycle structure The notation for the permutation operations can be abbreviated if we notice that the operations can be decomposed into cycles. For example, the permutation corresponding to C3 is composed of only the cycle (123). That is, vertex 1 maps into vertex 2 which in turn goes into vertex 3. Vertex 3 maps into vertex 1 which completes the cycle
2.2 The symmetric group
27
denoted by (123). The permutation corresponding to σ2 contains the two cycles (13) and (2). Thus, we could represent σ2 by the composition of its cycles (13)(2) or by just the single cycle (13), since a cycle of an object into itself is just an identity operation and can be dropped. The cycle structure becomes more apparent when we consider larger permutations, as illustrated with a few examples from the symmetric group of order 4, S4 : 1 2 3 4 1 2 3 4 = (1)(24)(3) ≡ (24), = (142)(3) ≡ (142), 1 4 3 2 4 1 3 2 1 3
2 4
3 2
4 = (1324), 1
1 4
2 3
3 2
4 = (14)(23). 1
(2.2)
Parity of a permutation A parity can be assigned to a permutation according to the length of its cycles. A cycle of odd length is assigned a parity of +1 and a cycle of even length is assigned a parity of −1. The parity of the permutation is the product of the parities of its cycles. Although assigning the parity in this way may at first be confusing, it is quite logical. A cycle of length one, for example, means the object is permuted into itself. There is no change and so the parity is +1. This definition of parity is equivalent to saying that the parity of an operation on a determinant is ±1 depending on whether the number of columns (or rows) interchanged is even or odd. Mathematica and combinatorics Mathematica provides a special package that handles permutations and other discrete mathematical functions. The permutation package can be loaded from Mathematica by the command
<< DiscreteMath ‘Combinatorica‘
The following Mathematica functions are useful:
• Permute[ l,p ] permutes list l according to permutation p. • InversePermutation[ p ] yields the multiplicative inverse of permutation p. • PermutationGroupQ[ l ] returns True if the list of permutations l forms a permutation group. • MultiplicationTable[ l,f ] constructs the complete multiplication table defined by the binary relation function f on the elements of the list l. For example f could be a permutation and the command becomes MultiplicationTable[l,Permute].
Symmetry and group theory
28
Mathematica provides the following function for cycle decomposition:
ToCycle[p], where p is a permutation.
Significance of permutations in computational algorithms The essential ingredients of computational group theory are that a permutation realization of a group is isomorphic to some other realization of the group (meaning they are equivalent, with a one-to-one relationship between group elements of the two realizations) and permutation groups are easy to manipulate computationally (meaning easy to manipulate on a computer). We now take a closer look at permutation operations, because of their central importance to computational group theory. The permutations of n objects form a permutation group known as the symmetric group, Sn . Since there are n! permutations of n objects, n! is the order of the group Sn . An example of a symmetric group is given by (2.1) for S3 , which is isomorphic to the group C3v .
2.2.1 The permutation group It is possible to describe every element of a finite group in terms of a permutation, as we have already seen. Thus, we can establish an isomorphism between the elements of a finite group G and a group of permutation, the latter satisfies the same multiplication table. This is just saying that we can always construct a permutation realization of any finite group. Such a group is not necessarily isomorphic with a symmetric group. We describe this permutation realization as a permutation group. One possible procedure for constructing a permutation group isomorphic to a finite group G is to make use of the rearrangement theorem. We learned that the operation of a group element X ∈ G, X = E, on the top row of the multiplication table simply rearranges the ordering of the elements in the row labeled by X. It permutes the elements. Thus, each element X ∈ G can be represented by the permutation defined by its row in the multiplication table.
Example 2.2 For the group C3v , we can e E= e e A= a e B= b
use Table 2.1 to define permutation a b c d f 1 2 3 4 = a b c d f 1 2 3 4 a b c d f 1 2 3 4 = e d f b c 2 1 5 6 a b c d f 1 2 3 4 = f e d c a 3 6 1 5
elements as follows: 5 6 , 5 6 5 6 , 3 4 5 6 , 4 2
2.2 The symmetric group e C= c e D= d e F = f
a d
b f
c e
d a
a c
b a
c b
d f
a b
b c
c a
d e
29
f b
1 = 4 f 1 = e 5 f 1 = d 6
2 5
3 6
4 1
5 2
2 4
3 2
4 3
5 6
2 3
3 4
4 2
5 1
6 , 3 6 , 1 6 . 5
In this case, there are six permutations, corresponding to the six elements of the group C3v . However, in the case of the symmetric group S6 , we would have 6!, or 720 permutation elements. Thus, the permutation group above is a subgroup of the symmetric group S6 . It is isomorphic to S3 .
Since a group is defined by its multiplication table, the first goal in our computational program is to construct it. Thus, rather than using the multiplication table to construct a permutation realization, we use the group elements, in permutation form, to construct the multiplication table. We begin by stating Cayley’s theorem, which ensures that there always exists a unique permutation group realization, but first we need to introduce the concept of subgroups. Subgroups Given a group G, it is often possible to select a subset of the elements of the group such that the subset H also constitutes a group under the same rules of binary composition, i.e. multiplication, as in G. H is called a subgroup of G, H ⊂ G. Of course, a group always has at least two subgroups, namely the group itself and the subgroup consisting of the identity alone. These trivial subgroups are called improper subgroups, all others being proper subgroups.
Example 2.3 For the group C3v , the elements C3 , C3 C3 = C32 , and C3 C32 = C33 = E form a cyclic subgroup, and C3 is of order 3. In terms of operations on the equilateral triangle, three successive applications of the operation C3 return the triangle to its original position. The set {E, C3 , C32 } forms a proper subgroup.
We are interested here in a group G that is a subgroup of a larger symmetric group Sn . The importance of the existence of such symmetric groups, Sn , is that we can assign a unique permutation of n objects to each element of G, a one-to-one mapping of all elements of G onto a subgroup of Sn . The result is known as Cayley’s theorem. Cayley’s theorem metric group Sn .
Every group G of order n is isomorphic to a subgroup of the sym-
This theorem asserts that there exists a finite number of distinct finite groups of order n, namely those found as subgroups of Sn .
30
Symmetry and group theory
It is clear from the above examples that the concept of a group and its binary multiplication relations transcends the particular physical realization. Thus an abstract group can be defined and identified according to the number of its elements and their multiplication relations. Corollary Two groups with an equal number of elements but different multiplication relations are distinctly different, otherwise they are isomorphic.
2.3 Computational aspects 2.3.1 Generation of the Cayley (multiplication) table Specification of a group: group generators and defining relations The binary operations defined on the set of group elements can be expressed in terms of a series of equations of the form Ai Aj = Ak ,
Ai , Aj , Ak ∈ G,
(2.3)
which in turn can be arranged in the form of a multiplication table (sometimes called a Cayley table) similar to the one displayed for the group C3v in Table 2.3. It is generally possible to also specify a group by using a smaller number of equations that imply the multiplication table. Such a set is called the defining relations for the group. The elements used in the defining relations are called the group generators. Rank, basis and presentation of a group The minimum set of group generators that can reproduce the multiplication table is called the group basis, which may not be unique. The number of generators present in the basis is called the rank of the group. The combination of defining relations and group generators is called a presentation of the group.
Example 2.4 For the group C3v , the group generators may be taken as σi and C3 , i.e. one of the σi . Then, the defining relations would be σi2 = C33 = E, C32 σi = σi C3 , and the rank of C3v is 2. We could just as well have chosen C32 instead of C3 .
Given a set of generators, the full group is obtained by operating on the generators with the aid of the defining relations. We should stress here that the generators of the isomorphic permutation group have the defining relations built in them. Thus, by taking powers of generators, products of generators, and products among newly generated elements, we obtain all the permutation group elements. The multiplication table is subsequently obtained.
2.3 Computational aspects
31
2.3.2 Computer generation of group elements Let us consider the C3v group, as an example. We take C3 and σ1 to be the group generators. In principle, it is desirable to use a group basis, i.e. a minimal set of generators, as input. In practice, a minimum set might not be immediately obvious, and one might choose to use an arbitrary set of group elements. The important thing is to generate the group. In general, one should know the order of the group and use it as input to the computer program. This will set the program to terminate element generation once the total number reaches the group order. First, we redefine the permutations of (2.1) in terms of a two-dimensional computer array N G(i, j): i is an index of the group elements, and we retain the index 1 for E, E ≡ 1. We follow that by setting the generator indices: C3 ≡ 2, σ1 ≡ 3, and so on. The index j assigns numbers to the objects to be permuted, three in the present case. The initial definition of the array N G(i, j) is as follows: N G(1, 1) = 1 E≡
N G(1, 2) = 2 N G(1, 3) = 3
N G(2, 1) = 2 C3 ≡
N G(2, 2) = 3 N G(2, 3) = 1
N G(3, 1) = 1 σ1 ≡
N G(3, 2) = 3 N G(3, 3) = 2
(2.4)
which follows from (2.1). Next we use these elements to generate the remaining three, with the help of the binary operation N G(m, j) = N G(i, j) × N G(k, j) = N G(i, N G(k, j)),
(2.5)
where N G(m) = N G(i) × N G(k).
Example 2.5 Let us find a fourth element of the group C3v by taking the group product C3 σ1 and denoting the result by N G(4) with array elements N G(4, 1), N G(4, 2), N G(4, 3). We will use (2.5), but first we illustrate it with an example. With a permutation realization for C3 and σ1 we have 1 2 3 1 2 3 1 2 3 C3 σ1 = = = σ3 . 2 3 1 1 3 2 2 1 3 We get this result by noting, for example, that 1 from the top box in the first column of σ1 goes into 1, which in turn goes into 2 from the first column of C3 , so the net result is 1 goes into 2 in the first column. In terms of (2.5) this is given by N G(4, 1) = N G(2, 1) × N G(3, 1) = N G(2, N G(3, 1)) = N G(2, 1) = 2.
Symmetry and group theory
32
Continuing in this fashion, we find
N G(4, 2) = N G(2, 2) × N G(3, 2) = N G(2, N G(3, 2)) = N G(2, 3) = 1, N G(4, 3) = N G(2, 3) × N G(3, 3)
(2.6)
= N G(2, N G(3, 3)) = N G(2, 2) = 3,
which corresponds to the permutations associated with σ3 in (2.1). The generation of the remaining elements is done in a similar fashion.
Given the group generators in array form, the number of permuted objects, and number of elements g in the group, a computer program can generate all other ments by successive multiplications of group elements till g distinct elements generated. The following is a simple Mathematica program for group generation; instead of array N G(i, j) it uses a list L, together with built-in functions we introduced above.
<
the eleare the
2.3 Computational aspects
33
This program can be expanded to include the construction of the Cayley table using the following Mathematica function:
Print["MULTIPLICATION TABLE"] m=TableForm[MultiplicationTable[L,Permute]] (∗ m is the multiplication table array. ∗)
For Fortran or C users, binary products of the elements of N G(i, j) should be carried out and the results stored in a two-dimensional array, say M U LT AB(i, j) = k,
(2.7)
where i, j, and k are the three group elements related by the binary operation. The generation of the Cayley table completes the usefulness of the permutation form of the elements. All subsequent derivations of group properties should be based on the generated Cayley table.
Example 2.6 In Example 2.5 we showed how to generate group elements from the generators of the group expressed as permutations. In this scheme the elements of the group are themselves expressed as numbers in an array. We found that with E ≡ N G(1) = 1, C3 ≡ N G(2) = 2 and σ1 ≡ N G(3) = 3 we generated a fourth element σ3 ≡ N G(4). If we had continued in this fashion, we would have found N G(5) = N G(2)×N G(4) and N G(6) = N G(2)×N G(2), which would have been represented in a Cayley table as
1 2 3 4 5 6
1
2
3
4
5
6
1 2 3 4 5 6
2 6 5 3 4 1
3 4 1 2 6 5
4 5 6 1 2 3
5 3 2 6 1 4
6 1 4 5 3 2
At this point, we would want to collect elements into sets of elements “similar” in nature, called classes. Classes play a central role in both abstract group theory and computational group theory. We will shortly show that C3 and C32 which are three-fold rotations, belong to the same class; they are labeled 2 and 6 in the above Cayley table, preferably, they should be relabeled 2 and 3. Consequently, we would prefer that the whole table be rearranged and that some elements be relabeled.
Symmetry and group theory
34
2.4 Classes The concepts of classes and class algebra are fundamental to both group theory and computational group theory. Classes consist of sets of group elements whose realizations are similar in nature. The symmetry operations σ1 , σ2 , σ3 ∈ C3v are all “similar” and belong to the same class. They all represent the same kind of operation. However, a rigorous definition is required because some elements that appear “similar in nature” can actually be different in a subtle way. (i) Conjugate elements mutually conjugate if
Two elements A and B belonging to a group G are called B = X A X −1
for some X ∈ G.
(2.8)
The operation on the r.h.s. is called conjugation. (ii) Classes A class is a set of all distinct mutually conjugate elements. Since the identity element commutes with all other group elements, we have XEX −1 = XX −1 = E, ∀ X. Thus E is in a class by itself, which we denote by C1 . Geometric interpretation There is a simple physical interpretation of conjugate elements when working with geometric realizations of a group. One can go from one element of the class to another with a similarity transformation (a conjugation!). To conform with the above definition of a class, the transformation, a matrix, must correspond to some element of the group. In a geometrical sense, say, it may rotate the coordinate system, so that the transformed operator maintains its original nature. Consider the geometric realization of C3v as the operations that leave the equilateral triangle unchanged. Then, for example, σ3 = C3−1 σ1 C3 has the following physical interpretation: A reflection σ3 through the C-axis of Figure 2.2 is equivalent to a 120◦ rotation by means of the group operation C3 , followed by a reflection through the fixed A-axis, and finally followed by a 240◦ clockwise rotation (120◦ counterclockwise) by means of the operation C32 = C3−1 , which returns the triangle to its original position. This means that two elements of a group are mutually conjugate if, in the geometric realization, they can be identified with equivalent physical or geometric operations (rotations through the same angle, for instance). We should stress, however, that the conjugation connecting the two operations has to be effected by another operation in the geometric realization of the group. If such a connecting operation is absent, the conjugacy of the two elements is also absent, despite their similar natures. In the above case, the equivalence of the elements C3 and C32 is clear. However, if we consider the symmetry operations of the square, shown in Figure 2.4, with point-group C4v , we find that not all of the nπ/2 rotations fall into one class. A C2 rotation is not connected by any of the elements of the group C4v to the C4 or C42 rotations. In the same manner, the reflections σv and σd belong to different classes for the same reason, they are not connected by any group element.
2.4 Classes
d
35
v
Fig. 2.4. The square mesh with C4v symmetry. Two inequivalent types of reflection planes are shown. Theorem A group G splits into “conjugacy classes” C1 , C2 , . . . , Cnc such that the following properties hold:
(i) Every element of G is in some class, and no element of G is in more than one class, so that G = C1 ∪ C2 ∪ · · · ∪ Cnc .
(2.9)
Sometimes this relationship is written with + signs rather than ∪ signs, but the relationship is a union of classes, not a sum of classes. (ii) All the elements in a given class are mutually conjugate and consequently have the same order s. (iii) An element that commutes with all elements of the group is in a class by itself and is called a self-conjugate element. The set of all self-conjugate elements of a group is called the center of the group. Because all the elements of an Abelian group mutually commute, each element is self-conjugate, i.e. in a class by itself. (iv) The number of elements ri in the class Ci is a divisor of the order of the group. Furthermore, the order of the elements of the class, si , is a divisor of g/ri . (v) Every product of two classes can be decomposed into a sum of classes: Ci Cj = hijk Ck ,
(2.10)
k
where the sum is an addition, as in ordinary arithmetic, not a union. This relationship is crucial for the development of computational group theory. In Section 2.4.2 we will elaborate on its significance in computational aspects of representation theory. (vi) The elements R of a class are rearranged by the conjugacy relation X −1 RX, just as the elements of a row or column of the multiplication table are rearranged by operating on every element by some member of the group other than the identity element. This is known as the class rearrangement theorem.
36
Symmetry and group theory 2.4.1 Computer generation of classes and class arrays
r In this section we introduce the definition of several Mathematica based functions that will be useful to subsequent programs. Using the definition of a conjugacy class as the distinct set of elements B = XAX −1 , ∀ X ∈ G, we next generate
(i) A one-dimensional array of inverse elements Inv(i) = j. In Mathematica this is implemented by the following code:
LInv=Map[InversePermutation,L] (∗ where LInv is a list of the inverses of the list L of group elements, given in the same order, and in permutation form. ∗) Print["INVERSE ELEMENTS IN PERMUTATION FORM: ",LInv] (∗ A compact form of the inverse elements is generated as follows: ∗) LInv={1} For[i=2,i
(ii) A two-dimensional integer array Class(i, j) = k identifying element k as the jth element in class i:
Class={{1}} i=1 nc=1 f:=m[[1,m[[1,j,i]],LInv[[j]]]] Block[{p=Range[g],C1={}},p[[1]]=0; While[Apply[Plus,p] =0,i=i+1; Switch[TrueQ[p[[i]] =0],True,C1={i};p[[i]]=0; For[j=2,j
2.4 Classes
37
AppendTo[Class,C1];nc=nc+1;C1={} ] ] ] Print["NUMBER OF CLASSES = ",nc] Print["CLASSES: ",Class] (iii) A one-dimensional array N CL(i), the number of elements in class i. In Mathematica the number of elements in class i is given by Length[Class[[i]]] (iv) the number of classes N C, given above as nc. (v) A one-dimensional array Ind Class (i), identifying the class to which element i belongs. In Mathematica this is generated as follows: Do[j=1; While[j
Do InvClass[i]=IndClass[LInv[[i]]],{i,1,g} (vii) A two-dimensional array IndPow (i, j), identifying the class to which ij belongs: Do
IndPow[i,1]=1;el=Class[[i,1;]];IndPOw[i,2]=i; Do[ Switch[i==1,True,IndPow[i,j]=1, False,el=m[[1,el,Class[[i,1]]]];k=1;
While FreeQ[Class[[k]],el],k=k+1 ; IndPow[i,j]=k ],{j,1,nn} ],{i,1,nc}
Symmetry and group theory
38
(viii) A one-dimensional array ClassExp(i) defining the order (or class exponent) of class i: ClassExp={1}
Do t=Class[[i,1]];ncc=1; While[t =1,t=m[[1,Class[[i,1]],t]];ncc=ncc+1]; AppendTo[ClassExp,ncc],{i,1,nc} Print["CLASS EXPONENTS: ",ClassExp]
2.4.2 Class multiplication We have asserted in (2.10) above that the product of two classes can be decomposed into a sum of classes Ci Cj =
hijk Ck ,
k
where classes may appear any number of times or not at all. This implies that we have to keep track of the number of times a class element appears. Class rearrangement theorem In order to prove this very important relation we will make use of the rearrangement theorem, which specifies that A conjugation operation simply rearranges the list of distinct elements of a class. Since a conjugacy operation may be considered in view of the associative property of group multiplication, (XA)X −1 , its outcome is unique as asserted by the group rearrangement theorem. Thus, given an arbitrary element X ∈ G, and two distinct elements A, B ∈ C; C ⊂ G, we assert that X A X −1 = X B X −1 , and hence X C X −1 = C,
∀ X ∈ G.
(2.11)
It is also true that any set of elements, S ⊂ G, satisfying X S X −1 = S, is composed of complete classes.
(2.12)
2.4 Classes
39
Class algebra To clarify what we mean by the product of classes, we begin by defining a sum over the elements of a class as ¯i = C A, (2.13) A∈Ci
where C¯i is called the class sum, and each element in the class appears only once in the summation. Consequently we can rewrite (2.9) as G¯ =
nc
¯i, C
(2.14)
i=1
where each class sum appears once and only once, and so each group element appears once and only once. However, our new introduction of the operation of addition of group elements allows to count frequent occurrences of an element, for example we write A + A = 2A ! This is more akin to traditional arithmetic and the ordinary meaning of summation. By introducing this extra binary operation, to go along with the binary composition of multiplication of group elements, we have defined an algebra on the group, the Frobenius algebra. We can use (2.11) to prove (2.10) ¯iC ¯j = X C ¯ i X −1 X C ¯ j X −1 C ¯ j X −1 , ¯i C = XC ¯ j , which demonstrates that C ¯ j is composed of ¯iC ¯iC where we also used (2.12) with S = C complete classes, and that a given class may appear more than once. ¯iC ¯ j can be decomposed into a sum of classes, Thus, every product of two class sums C ¯j = ¯iC C
nc
¯k , hijk C
(2.15)
k=1
where the coefficients hijk are called the class multiplication coefficients, and are positive integers or zero. This equation is frequently written without the bar over the class symbols and is described simply as a product of classes rather than a product of class sums.
Example 2.7 For the group C3v we can define the class sums ¯ 1 = E, C
¯ 2 = C3 + C32 ; C
¯ 3 = σ1 + σ2 + σ3 . C
¯i = C ¯1 = C ¯ 3 = 2C ¯2, ¯1C ¯iC ¯i, C ¯ 22 = 2C ¯1 + C ¯2, C ¯2C ¯3 = C ¯3C The class multiplications are then: C 2 ¯ ¯ ¯ C3 = 3(C1 + C2 ).
Symmetry and group theory
40
Theorem A class sum of a group G commutes with all the elements of G. Proof For A ∈ G we define a conjugacy class of A as ¯ RAR−1 . C(A) =
(2.16)
R∈G
Then, for any element X ∈ G, we find ¯ X C(A) =
XRAR−1 X −1 X
R∈G
=
(XR)A(XR)−1 X
R∈G
¯ = C(A)X.
(2.17)
Since a class sum commutes with all elements of the group, it must commute with all other classes. Thus Ci Cj = Cj Ci , and so from (2.11) we must have hijk = hjik .
(2.18)
2.4.3 Computer generation of class multiplication matrices The class multiplication coefficients hijk are defined as a three-dimensional array H(i, j, k). It is generated by enumerating the frequency of occurrence of a representative element of class k when all elements of class i are multiplied by all elements of class j.
Example 2.8 Using Example 2.7 we have ¯ 3 = (C3 + C32 )(σ1 + σ2 + σ3 ) = 2(σ1 + σ2 + σ3 ). ¯2C C ¯ 3 in order We see that we need only enumerate the occurrence of, say, σ1 from the class C to determine h233 = 2.
In Mathematica we generate the class multiplication matrices with the aid of the following code:
Print["CLASS MULTIPLICATION MATRICES"] Do[ H[i,j,k]=0,{i,1,nc},{j,1,nc},{k,1,nc}] Do[H[1,i,i]=1;H[i,1,i]=1,{i,1,nc}]; s:=m[[1,Class[[i,l]],Class[[j,k]]]]; Do[ For[l=1,l
2.5 Homomorphism, isomorphism, and automorphism
41
For[k=1,k
2.5 Homomorphism, isomorphism, and automorphism Given two groups G and G , a mapping θ of G onto G that preserves multiplication is called a homomorphism. For a homomorphism θ, (θAi )(θAj ) = θ(Ai Aj )
for all Ai , Aj ∈ G.
(2.19)
The smaller group is called the homomorph of the larger. The subset of elements of G that map onto the identity of the smaller group G is called the kernel of the homomorphism. • If θ is a one-to-one mapping, it is called an isomorphism and the groups G and G are said to be isomorphic. • If θ is an isomorphism and G = G , then θ is called an automorphism. • If θ is an automorphism and the mapping from G onto itself is such that θ G = X G X−1 ; for X ∈ G, that is, such that the automorphism is produced by conjugation, it is called an inner automorphism, all others being called outer automorphisms. Examples (i) If G = G6 and G = G2 ≡ {E, A }, then the mapping θ(E) = θ(D) = θ(F ) = E θ(A) = θ(B) = θ(C) = A is a homomorphism of G6 onto G2 .
Symmetry and group theory
42
(ii) The groups C3v , D3 and S3 are isomorphic: They have the same order and share the same multiplication table. (iii) If θ(E) = E, θ(A) = B, (B) = C, θ(C) = A, θ(D) = F , and θ(F ) = D, the mapping is an automorphism of G6 onto itself.
2.6 Direct- or outer-product groups Given two groups G and H of order g and h, such that all the elements of G commute with all the elements of H, Gi Hj = Hj Gi
∀Gi ∈ G and Hi ∈ H,
(2.20)
Table 2.4. Schoenflies (Sch.) and Hermann–Mauguin (H-M) notation for point-groups. Group notation Sch.
Group
H-M
Cn
n
Cnv
nmm nm
S2n
n ¯
Cnh
n m n ¯
Dn
order
Group generators
n
n
Cn
nmm nm
2n 2n
Cn , σv , σd Cn , σv
n ¯
2n
S2n
2n
Cn , σh
(n odd)
n m n ¯
2n
Cn , I
n22 n2
(n even) (n odd)
n22 n2
2n 2n
Cn , C2 , C2 Cn , C2
Dnh Dnd
n 2 2 mmm n ¯m
(n even) (n odd)
n mmm n ¯m
4n 4n
Cn , C2 , C2 , σh Cn , C2 , I
T
23
23
12
C3 , C2
Th
2¯ 3 m
m¯ 3
24
C3 , C2 , I
Td
¯ 43m
¯ 43m
24
C3 , C2 , σv
O
432 4¯2 3 m m
432
24
C4 , C3 , C2
m¯ 3m
48
C4 , C3 , C2
Oh
(n even) (n odd)
(n even)
Column 2 gives the full H-M notation, while column 4 gives the abbreviated one.
Exercises
43
and the only element in common is the identity, then the set of all combinations Gi Hj forms a group of order gh, called a direct- or outer-product group. Product groups, including the direct-product, are the subject of Chapter 8. Here we simply note that the importance of direct-product groups often arises when the two sets of operators (those from group G and those from group H) operate on independent sets of functions or coordinates. Examples are given in Exercises 2.17–2.19. The meaning of (2.20) is illustrated as follows. Given a group GA with i elements (E, A2 , A3 , . . . , Ai ) and another group GB with j elements (E, B2 , B3 , . . . , Bj ), the directproduct group is (E, A2 , A3 , . . . , Ai ) ⊗ (E, B2 , B3 , . . . , Bj ) = (E, A2 , A3 , . . . , Ai , A2 , A2 B2 , . . . , Ai B2 , . . . , Ai Bj ),
(2.21)
which forms a group containing ij elements.
Exercises Note on the exercises: Exercises 2.1 through 2.17 range from those which help in developing an understanding of the theory of groups to those which are in the nature of finger exercises and help in developing familiarity with group theory and some dexterity in performing the mathematical manipulations of group theory. Exercises 2.17 and 2.18 are crucial. The solution to Exercise 2.18 provides the basis for the remaining computational methods that follow in later chapters. Exercise 2.19 provides a check on the program developed in Exercise 2.18. Exercises 2.20 through 2.25 provide an introduction to crystallographic point-groups. They should all be read and thought about, and at least a few of them carried to completion. Geometric figures are provided to elucidate the properties of these point-groups. The vertices of the figures are numbered sequentially, to facilitate the construction of permutation operations associated with the groups. In addition to the particular questions posed in each problem, apply the program developed in Exercise 2.18 to each of Exercise 2.20 through 2.25. 2.1 Convert the following permutation from bracket notation to cycle notation: 1 2 3 4 5 6 7 6 5 2 3 4 1 7 2.2 Convert (163275) to bracket notation. 2.3 Given permutation operators p and q defined by 1 2 3 4 5 6 7 1 p= q= 1 7 4 5 3 2 6 5
2 1
3 2
4 4
5 3
6 7
7 , 6
(a) find the product pq in bracket notation; (b) use the Permute function defined in Mathematica, or any other computer language code you develop, to carry out the permutation product. 2.4 Repeat Exercise 2.3 for the following pairs of permutation operators: (a) p = (567), q = (2673); (b) p = (246)(37), q = (143)(56).
Symmetry and group theory
44
Write the products pq in cycle notation. Try to do this by sight without writing out the implicit cycles in p or q. 2.5 Find the inverse and degree of each of the following permutation operations, by longhand, using the Mathematica function InversePermutation, or developing your own code in C or FORTRAN: 1 2 3 1 2 3 1 2 3 4 p= q= r= 1 3 2 3 2 1 4 3 1 2 1 2
s=
2 1
4 3
3 4
u=
1 1
2 4
3 7
4 6
5 3
6 2
7 5
t=
1 5
2 4
3 1
5 3
4 2
v=
1 6
2 5
3 2
4 3
5 4
6 7
7 1
w = (12)(34657) 2.6 Show that the permutations of n objects which form the symmetric group Sn is of order n!. 2.7 Show for the symmetric group S3 that elements with the same form of decomposition into cycles belong to the same class. When generating the group classes with a Mathematica program, use the function ToCycle[p] to generate the cycle structure of the elements. 2.8 Determine the classes of the symmetric group S4 . 2.9 Show that the number of elements nc(i) of a class Ci of a finite group G divides its order, i.e. g/nc(i) is an integer. 2.10 Show that the set composed of all inverses of the elements of a class Ci of a group G is also a class of G, which we may denote by Cj = C−1 i . Such classes are called mutually reciprocal classes. If a class contains its own inverse elements it is called a self-inverse class. 2.11 Consider the isomorphic realizations C4v and D4 of the square. These realization groups contain 8 elements: E, C4 , C4−1 , C2 , σ1 (C21 ), σ2 (C22 ), σ1 (C22 ), σ2 (C22 ). In addition to the identity operation, we find in each realization four-fold rotations and reflections (or two-fold rotations). However, if we examine the class structure of these realization groups, we find: C1 = {E}, C2 = {C4 , C4−1 }, C3 = C2 , C4 = {σ1 (C21 ), σ2 (C22 )}, C5 = {σ1 (C22 ), σ2 (C22 )}. How do you justify this class structure when each of the set of reflections or the set of two-fold rotations are similar in nature? 2.12 Prove that if class Cj contains the inverse of element R in class Ci , then Cj must be composed of all the inverse elements of Ci , and nc(j) = nc(i), where nc(i) is the number of elements in class Ci . An ambivalent class is a class that is its own inverse. 2.13 Find the class multiplication coefficients hijk for the groups C3v and C4v . 2.14 Show that the following general relations are satisfied by the class multiplication coefficients.
Exercises
45
(i) hijk = hjik (This is equivalent to proving that Ci X = XCi for all elements X. Let X range over all elements in Cj .) ncl ncl (ii) hjlk hikm k=1 hijk hklm = ncl k=1 (iii) nc(i)nc(j) = h nc(k). ijk k=1 (iv) hijk = h¯i¯j k¯ (v) nc(k)hijk = nc(i)hk¯ji = nc(i)hj k¯¯i = nc(j)hik¯¯j (vi) hij1 = nc(i)δi¯j where nc(i) is the number of elements in class Ci and where bars denote the inverse class. That is, C¯i is a class that contains the inverses of the elements of class Ci . Note that the third subscript on h is the number 1, not the letter l. (vii) Show that a mapping of one group onto another can be completely specified by the action of the mapping on the generators of the larger group. 2.15 Prove the group rearrangement theorem. 2.16 Prove the class rearrangement theorem. 2.17 Prove that the set of integers 1, 2, 3, . . . , (k − 1) form a group of order (k − 1) under ordinary multiplication modulo k. (Note: Two integers m and n are equal, modulo k, if m = n + jk, where j is an integer.) Multiplication, modulo a prime number, plays an important role in Dixon’s method for determining the characters of irreducible representations. 2.18 Write a general computer program, guided by the outlines in the text, which makes use of the minimal set of group generators to (i) (ii) (iii) (iv) (v)
generate the group elements in permutation form, construct the corresponding Cayley tables, generate the inverse elements, generate the classes and class arrays, specified in Section 2.3.1, generate the class multiplication matrices.
2.19 Use the program from the previous problem to obtain the group multiplication tables for the point-groups C3v , C4v , C5v . The surface nets of Figure 2.5 can be modified by replacing the reflection planes σv with two-fold rotation axes C2 that are perpendicular to the principal Cn axis, and by replacing the reflection planes σd with two-fold C2 axes, giving rise to the dihedral symmetry groups Dn shown in Figure 2.6, which are isomorphic to the Cnv groups. 2.20 Figure 2.5 shows the primitive meshes corresponding to allowed two-dimensional surface lattices (nets). The vertices are sequentially numbered, clockwise. Also shown are the allowed types of reflection planes, designated by σv and σd . (a) Find all the physically realizable point-symmetry operations for the four meshes of Figure 2.5. Write out these symmetry operations as permutations of the vertex numbering, in cycle notation. (Note: there are 4, 8, 6, and 12 operations for these meshes, respectively; and the identity operation and the rotations maintain the clockwise ordering of the labeling. The mirror reflections change the labeling to counterclockwise.) (b) Why aren’t the remaining permutations, like (1324) for C4v symmetry operations?
Symmetry and group theory
46 1
2
1
2
sv 4
sv
3
3
4sd
sv 1
sv
2
1 6
3
2
3
5
4
Fig. 2.5. Clockwise from top left: symmetries of the point-groups C2v , C4v , C6v , and C3v , respectively. C2
C4
C2
C2
C C2
C29
2
4
C3
C6 C2
C2
C29
C2 3
C29 C2
6
C C2
Fig. 2.6. Symmetries of the dihedral groups D2 , D4 , D3 , and D6 .
2.21 Figure 2.7 shows the primitive (Wigner–Seitz) cells for lattices with symmetry involving a major axis of rotation and a horizontal reflection plane σh , that is, a reflection plane perpendicular to the major axis. These improper point-symmetry groups are designated Cnh , where n = 2, 3, 4, 6. Show that these groups have the following properties: n/2
(i) Since groups with even n include the two-fold rotation C2 = Cn , by taking the major axis along the z-direction, and defining the C2 and σh by the threedimensional rotation matrices, show that ⎞ ⎛ −1 0 0 ⎟ ⎜ C2 σh = σh C2 = ⎝ 0 −1 0 ⎠ = I, 0 0 −1
Exercises
47
2
2
3
1 4
sv
1
sh
3 4
6
7
6 7
5 8
sd
5
8
Fig. 2.7. Primitive cells with Cnh symmetries. which is just the matrix that defines the inversion symmetry operation r → −r. Thus, Cnh symmetries with even n contain the inversion operation, i.e. the corresponding primitive cells have a center of inversion. Groups with odd n do not contain the inversion. (ii) For n = 1, the group C1h is composed of the identity E and σ, and is usually denoted by Cs . Thus, show that we can express the Cnh groups as the outerproducts Cnh = Cn ⊗ Cs , containing 2n elements. (iii) Each has 2n classes. 2.22 When the dihedral groups Dn are augmented by a σh reflection plane, perpendicular to the major axis, as shown in Figure 2.8, we obtain the improper point-groups Dnh . Again, I is an element of the group, only if n is even. Show that the Dn point-groups have the following properties: (i) The group order is 4n. (ii) They contain n σv reflection planes, in addition to the n C2 rotations. (iii) σh commutes with all the elements of the group. Hence we can express these groups as Dnh = Dn ⊗ Cs . 2.23 In Figure 2.9, we show the case of augmenting Dn by a vertical σd reflection plane that bisects the angle between two neighboring C2 axes. The ensuing groups are designated Dnd . Show that (i) the operation C2 σd = S2n , where C2 is one of the neighboring two-fold axes;
Symmetry and group theory
48
2
2 1
3
3
1 4
4 sh
6
6
7
7
5
5
8
8
D2h
D4h 2
2 3
4
1
5 6
3
1 6
5
8
9 10
7 12
4
11
D3h
D6h
Fig. 2.8. Primitive cells with Dnh symmetries. 2 2
3
3 1 4
1
5
6
6
7
5 4
D3d
8 D2d
Fig. 2.9. Primitive cells with Dnd symmetries.
Exercises C3
49 C3
1
sd
C2
C2
2
4
4
3 T
3 Td
Fig. 2.10. Primitive cells with tetrahedral symmetries T and Td .
(ii) for n odd, there is one σd plane perpendicular to one of the two-fold axes, and that, in this case, the group can be expressed as Dnd = Dn ⊗ Ci . 2.24 Figure 2.10 shows two regular tetrahedra with symmetries T and Td . (a) For the tetrahedron shown with point-group symmetry T , write out the symmetry operations in cycle notation for the various rotations about the axes that pass through an apex of the tetrahedron and the center of the opposite face. These consist of rotations denoted by C3 and C32 . Do the same for symmetry operations that consist of rotations about a two-fold axis that passes through the midpoint of one edge, the center of the tetrahedron, and the midpoint of the opposite edge. Show that these 11 operations together with the identity form a group, the T group. (b) In addition to the operation of part (a), the tetrahedron with point-group symmetry Td shows reflection planes that pass through one edge of the tetrahedron and bisect the opposite edge. Each of these reflection planes contains one 2-fold and two 3-fold axes, and bisects the angle between the remaining two 2-fold axes, thus designated σd . Show that each σd plane converts the 2-fold axis it contains into a 4-fold rotary reflection axis S4 . Write out the symmetry operations corresponding to the reflections σd in cycle form. Expand the group of part (a) by including these symmetry operations in the group. Note that this requires the inclusion of other symmetry operations to complete the group, an example being (1234) = (14)(123), which corresponds to a rotation followed by a reflection. This group is designated Td . (c) Figure 2.11 shows the primitive cell with Th symmetry. In this figure the threefold axes are rotated to coincide with the body diagonals of a cube. One of the two-fold axes is now along the z-axis. The σh reflection planes are perpendicular to the two-fold axes and bisect the angles between the three-fold axes. Carry out all the steps stated in parts (a) and (b).
Symmetry and group theory
50
C3 [111]
C2
1 2 sh
3 4
Fig. 2.11. Primitive cell with Th tetrahedral symmetries. C3 [111]
C3 [111]
C4
C4
1
5 4 1 3
2 3
4 C2 6
sh
2 C2 7
6 5
8
Fig. 2.12. The octahedral primitive cell with O and Oh symmetries. 2.25 Figure 2.12 shows the primitive cell with O and Oh symmetry. O is composed of allowed rotation and reflection operations except σh , it has 24 operations. Obviously, Oh contains σh . Carry out all the steps stated in parts (a) and (c) of the previous problem for these two octahedral groups.
3 Group representations: concepts
In Chapter 1 we presented configuration-space translation operators, tA , and functionspace counterparts, " tA . The latter transforms functions such that the value of the new transformed function at the new transformed coordinates equals the value of the original function at the original coordinates. Generalizing these operators to include rotations and other transformations, we write the defining relation between configuration-space " as operators R and function-space operators R " (r) = f (R−1 r). Rf
(3.1)
Note carefully here that we refer to two different spaces. We have configuration space, in which the operator R is defined, and function-space in which the function operator " is defined. The latter effects transformations among functions in Hilbert (function) R space. A key point A coordinate transformation in configuration space induces a transformation of a function defined on the space. Equation (3.1) defines the way in which we determine the function-space operator. In Chapter 2 we discussed configuration-space operators that provide a geometric realization of a group. We used permutation matrices to describe mathematically the effect of configuration-space operators on objects placed in that space. These permutations obey the group multiplication table and are also a realization of the group. We employed them to generate the multiplication table of the group. Generally, however, we find it more useful to use configuration-space matrices that operate on vectors. Such matrices are the natural choice for describing symmetry properties related to geometry, such as translations, rotations, and reflections. This set of matrices also obeys the group multiplication table and is also called a realization of the group. We now introduce the concept of a representation of a group. A tentative definition A representation of a group is a mapping of the elements of a group onto a set of matrices that obey the group multiplication table.
51
52
Group representations: concepts
The reader will note that a few sentences earlier we stated that the set of configurationspace matrices that obey the group multiplication table is a realization of the group. Here, we should remind the reader that the realization is required to be isomorphic, i.e. we have to establish a one-to-one correspondence between distinct group elements and distinct matrices, and ensure that this mapping satisfies the multiplication table. Thus, we identify a matrix realization of a group as a representation, but we should be careful and state that not all representations can be accepted as realizations. For example, we may map all group operators onto the identity matrix. This satisfies the multiplication table, but cannot be a realization. Thus, we mean much more by group representations than just geometrical transformations in configuration space. Having introduced the more general concept of a representation of a group, we are now " may induce a representation of a group G in set to explore how function operators R function-space, i.e. how to construct matrices that effect transformations induced by the " on a set of functions. To find the effect of a function operator acting on a function, we Rs must employ (3.1). This is both important and nontrivial. It is often the source of much confusion. What we mean by a representation will become clear after some further development of the idea, helped along by examples. If physical applications of group theory were simply a matter of rotating triangles and the like, the subject would be trivial. It is the application to functions that makes all the difference between triviality and substance. Representations are central to the basic theory, and so it is important to develop a clear understanding of the main concepts, particularly what a representation is and how it may differ from a realization.
3.1 Representations and realizations As a prelude to matrix representations, and as a way of developing a connection between permutation matrices and matrix descriptions of coordinate transformations, we now consider permutations of three balls of different colors: red, blue, and green. Following our prescription for realizing a group, we have: (i) objects: three balls; (ii) characteristic: color (red, blue, and green); (iii) operations: permutations of the balls. Using the permutations defined in (2.1), with 1, 2, and 3 replaced by red, blue, and green, we can construct the realization given in Figure 3.1. Figure 3.1 is simply a pictorial way of describing the permutations of the colored balls; it can be transformed into a matrix format as shown in Figure 3.2, which is a kind of half-way house between the description provided by (2.1) and the matrix realization that results from replacing all dots by ones and placing zeroes elsewhere. Now suppose the balls were all the same color. There would be no distinguishable configurations, thus all permutations could be realized by the identity matrix or by just the number 1. We will see, in following chapters, that this is an important representation of the group even though it is trivial here.
3.1 Representations and realizations R
B
G
R
E
G
B
G
B
A
R
B
B
B
G
53
R
G
C
R
G
D
R
B
F
Fig. 3.1. A colorful representation of the permutations of (2.1).
Red Blu
Grn
Red Blu
Grn
Red Blu
Grn
Red Blu Grn E
A
B
C
D
F
Red Blu Grn
Fig. 3.2. A permutation-matrix representation of G6 .
The representation for the permutations of colored balls displayed in Figure 3.1 could also serve as a realization for the set of operations that leave an equilateral triangle unchanged. This could be done simply by letting the colors of the balls represent the vertices of the triangle. On the other hand, if the object were a sphere and the operations
Group representations: concepts
54
were those that leave the sphere unchanged, the result would be the same as for the case with all balls the same color.
3.1.1 Transformation of coordinates We have seen that the symmetry operations of the equilateral triangle obey the group multiplication rules of Table 2.2. It is physically easy to perform these operations with a cardboard cutout of an equilateral triangle or even simply to imagine them. Rotating or inverting the triangle is quite natural. One knows that it can be done. The question is, “How does one describe these operations mathematically?” To start with, we need a reference frame. For permutation groups, the frame of reference is the boxes among which we permute objects. For rotation groups it is a set of coordinate axes. Suppose we consider the clockwise rotation of the equilateral triangle as depicted in Figure 2.3. We use a right-handed coordinate system with fixed axes (x, y, z). The z-axis points out from the plane of the triangle. The element C3 rotates the triangle clockwise by 120◦ as viewed from above the plane of the triangle. Vertex 1, which originally lay on the positive y-axis, now lies below the x-axis. To describe this rotation of the triangle mathematically, we do just as we did in Chapter 1 for a traveling wave: we refer to the inverse problem. We recognize that rotating the triangle clockwise in a fixed coordinate system is mathematically equivalent to keeping the triangle fixed in space and rotating the coordinate system counterclockwise, as illustrated in Figure 3.3. Then, under the passive operation C3 , denoted by C3 (p) in Figure 3.3, the new coordinates (x , y ) in the new (x , y ) coordinate axes are given in terms of the original coordinates (x, y) by x y
=
cos
2π
− sin
sin
3
2π
cos
3
2π
3
2π = 3
− 12
−
√ 3 2
√
3 2
− 12
x
.
(3.2)
y
y 1
1
x9
C3(p)
x 3
3
2
2
y9
Fig. 3.3. A 120◦ counterclockwise rotation of the coordinate system by the passive operator C3 (p)to a new (x , y , z ) coordinate system.
3.1 Representations and realizations Thus the operation C3 can be described mathematically by the matrix √ 1 3 −2 2 , C3 = √ − 23 − 12
55
(3.3)
but in the active view we interpret C3 to describe the clockwise rotation of the triangle by 120◦ about the origin of a fixed coordinate system, as was done in Chapter 2, Figure 2.3. Examining all operations of the group C3v in this manner, we find the following set of matrices to describe the operations. √ 1 − 23 −1 0 1 0 2 E= σ2 = σ1 = √ 0 1 0 1 − 23 − 12 (3.4) √ √ √ 1 1 1 3 3 −2 − 2 − 23 2 2 2 2 σ3 = √ C3 = C3 = √ √ 3 3 − 12 − 12 − 23 − 12 2 2 We enclose the matrices in brackets [· · · ] to indicate that they correspond to transformations of coordinates in the physical space in which symmetry operations on the real objects take place. These sets of matrices are mathematical realizations of the group. We refer to the transformations as taking place in configuration space. We reserve brackets (· · · ) to enclose matrices that represent operations in function-space. Equations (3.4) provide a matrix realization of the group G6 . They are often called a matrix representation of C3v , but they are not the representation, nor are they a representation in the sense of representing operations on functions. We generally reserve the term representation for those sets of matrices that represent groups of function operators, a subtlety we will deal with in more detail. The group of physical symmetry operations that transform an object into an equivalent position, in a fixed coordinate system, is useful for generating a group multiplication table. That is how we generated Table 2.2, which represents a physical realization of the group. These symmetry transformations are directly related to the physical symmetry of the system; but they may not include all the symmetry transformations of its Hamiltonian, exchange of electrons being an example of a symmetry not included.
3.1.2 Transformation of functions Our primary interest is the group of symmetry transformations of the Hamiltonian and operations on functions that are eigenfunctions of the Hamiltonian. For these purposes, we treat the mathematical description of coordinate transformations (configuration-space operators) as a means of generating representations. Matrix realizations such as (3.4) are also a representation in some sense but not in the sense of representing operations on functions. The distinction between realizations and representations is important if one is not to miss the whole point of generating representations. Our goal is to make this clear and thus avoid a common misperception. Both group operators in configuration space and group operators in function-space obey the group multiplication table. The two groups are isomorphic, subject to a degree of subtlety to be described further on in this section; but the group of function operators
56
Group representations: concepts
is closer to our real interests, which, for example, may be eigenvalues and eigenfunctions. Transformations in configuration space provide a means for inducing transformations in function-space, as we illustrate in this section. " in contrast to Function operators, or F-operators, are represented by the symbol R, their configuration counterparts, which are given simply the symbol R. The relationship between the two sets of group operators is formally defined by the equation " (Rr) = f (r) Rf
(3.5)
or its equivalent, but generally more useful, form
" f (r) = f (R−1 r), R
(3.6)
" f is itself a function. Thus the value of the function f at the point r which shows that R " at the transformed point r = Rr. equals the value of the new (transformed) function Rf " f (r) as a new function, f (r), defined in the same coordinate system as We interpret R " That is, f (r), and obtained by operating on f (r) with an F-operator R.
" f (r) = f (R−1 r). f (r) = R
(3.7)
Equation (3.7) provides a simple but complete prescription for determining the new " (r), given f (r). function Rf A prescription To generate a new function from a given function f (r), replace the components of r in f (r) by the corresponding components of R−1 r. It is through this prescription we will find matrix representations of function operators that satisfy the group multiplication table. We denote operations in configuration space as before, with symbols that indicate the kind of symmetry operation, such as C3 for a clockwise rotation by 2π/3 about a three-fold symmetry axis; but, for the corresponding operations in function-space we include a caret over the operator to indicate that it operates on functions, in conformity with (3.6). For "3 to denote a function operator corresponding to the configuration-space example, we use C operator C3 . "3 , corresponding to (3.6) is The relationship between operators C3 and C ˜3 r), "3 f (r) = f ((C3 )−1 r) = f (C C
(3.8)
where the second equality holds because (C3 )−1 is a real orthogonal matrix and thus ˜3 . Generally, we deal only with real orthogonal matrices, equals the transpose matrix C these being the kind of matrices that describe the transformations between orthogonal coordinate systems, or with unitary matrices.
3.1 Representations and realizations y
57 y
^
C3 ⫹
⫺
x
⫺
x ⫹ 1 3 p ⫺ px ⫺ 2 2 y
px
"3 acting on px produces − 1 px − Fig. 3.4. C 2
√ 3 p . 2 y
In the active picture, which we have chosen, C3 is a configuration-space operator that "3 , which carries a point clockwise by 120◦ (Figure 2.3) and induces the function operator C ◦ "3 on “carries” a function clockwise by 120 , as illustrated in Figure 3.4. The action of C the function is determined by (3.8) and it is represented by the matrix for C3 , as we will soon see. Now, for a simple but important illustration, let the functions of interest be atomic p-states px = xe−r ,
py = ye−r ,
pz = ze−r .
(3.9)
We can ignore the exponential factor, when convenient to do so, since all the directional dependence in these functions is contained in the prefactor. Considering pz , it is clear that none of the operations in the group C3v change the z-dependence. Thus we must have, for example, "3 pz = ze−r = 1 × pz . C
(3.10)
Since all operators of the group C3v leave pz unchanged, we have generated a representation of the group in which all the elements are mapped onto unity. This is a homomorphic mapping, that is, a many-to-one mapping which satisfies the group criteria as well as the multiplication table for C3v . It is a representation because it describes the effect of group operators on functions, the p-orbitals. We could do the same thing with a vector or a dipole along the z-axis and obtain the same result. In general, whether we are dealing with a vector, a dipole or a p-orbital, the representation will be the same. The representation transcends the physical system and is characteristic only of the symmetry properties. We have just encountered an important difference between realizations and representations. The set of configuration operators constitute a realization of a group. In this case the operators {E, C3 , . . .} are a realization of the group C3v , and they satisfy the multiplica" C "3 , . . .} also satisfy the multiplication tion table for the group. The set of F-operators {E, table for the group C3v , and this set of operators is isomorphic to the configuration-space operators, as we proved in Section 1.3. However, and this is the important difference, the
58
Group representations: concepts
set of matrices that describe operations in configuration space are all distinct. There is a unique matrix for each configuration operator and, in some sense, one could say the matrices are the operators; but there is no such restriction for the matrices that represent function-space operators. We have just encountered a case in which the entire set of function-space operators are “represented” by 1 × 1 matrices, all of which are equal to unity. Now let us consider the effect of group operations on px and py . From (3.8), substituting px for f , we have "3 px (r) = px (C3−1 r), C
(3.11)
"3 py (r) = py (C3−1 r). C
(3.12)
and, similarly,
Using the inverse of C3 , which is the transpose of C3 , we find √ √ 1 − 12 − 23 x − 23 y x − −1 2 √ √ C3 r = = . 3 3 y x − 12 y − 12 2 2
(3.13)
Note that in (3.13) we have used brackets [· · · ], because these are operations in configuration space. "3 operating on a p-orbital just replaces r by Since in (3.11) and (3.12) the operator C −1 C3 r, we see from (3.13) that x is replaced by √ 3 1 y − x− 2 2 and y is replaced by √
1 3 x − y. 2 2
Thus, √ "3 px = − 1 px − 3 py , C 2 2 √ 1 3 "3 py = p x − py . C 2 2
(3.14) (3.15)
Since in both cases we obtain linear combinations of px and py , we can treat these two "3 and describe the operation of C "3 on functions as partners under the transformation C these functions by the equation "3 C
1 −2 px = √ 3 py 2
−
√
3 2
− 12
px . py
(3.16)
Now (3.16) is mathematically correct, and it is possible to write similar equations for the remaining function operators of the group, but they are not a good choice. The problem
3.1 Representations and realizations
59
with this choice of matrices is that they do not satisfy the group multiplication table. (See Exercise 3.3) That is why we have used brackets [· · · ]. The set of 2×2 matrices determined as was done for (3.16) is not a representation of the group. The proper way to express the relationship of (3.16) is √ 3 −1 2 2 " C3 px py = px py , (3.17) √ − 23 − 12 "3 . That is, in order where the 2 × 2 matrix represents the action of the function operator C to use the coordinate transformation matrix as a representation, we must treat px and py as components of a row matrix and operate, from the left, on the transpose of the 2 × 2 matrix of (3.16). With this choice we obtain (3.17). We also have consistency, since then the matrices used for realizations conform with those for representations. We can write (3.17) in more general form as " Ψi = " R Ψj Γji (R), (3.18) j
where Γji are the elements of a matrix representation. We will prove the correctness of this more general form in Chapter 4. "3 The operator C3 rotates the object clockwise. C "3 creates Comparison of C3 and C a new function that “looks” like the original function but has a different functional form. Put differently, C3 rotates the triangle, or NH3 molecule, carrying with it the original numerical value of px but expressed after rotation in a new functional form as given by "3 px . See Figures 2.3 and 3.3 for comparison. C We could continue in this fashion for all the function-space operators for the group C3v and determine the set of matrices that constitute a matrix representation for C3v . The result is another representation as given by (3.19). 1 0 −1 0 E= σ1 = 0 1 0 1 √ √ 1 1 3 − 23 2 2 2 σ2 = σ3 = √ √ (3.19) 3 − 23 − 12 − 12 2 √ √ 1 1 3 −2 − 2 − 23 2 2 C3 = C3 = √ √ 3 − 23 − 12 − 12 2 We notice, perhaps not surprisingly, that the representation is just the set of matrices describing the rotation of a vector in two dimensions by the group C3v , and that the set of matrices given by (3.19) satisfies a group multiplication table for the group C3v that is isomorphic to Table 2.2. This is, as stated, perhaps not surprising, but it can also be the source of confusion. One source of confusion is the common practice of illustrating function operators by using p-orbitals for the functions, just as we have done. The potential source of confusion is the vector nature of p-orbitals. How does one distinguish a vector from a function? It appears as if we have gained nothing. We have, however, learned that representations need not be a set of unique matrices; later we will
Group representations: concepts
60
deal with functions of higher order, which should clarify further the nature of function operators. For now, we have found two representations for C3v based on operators in functionspace. One of these is a set of operators all of which are equal to unity. The other is the set of 2 × 2 matrices given by (3.19). We can incorporate both of these representations into a single 3 × 3 matrix representation for which all the matrices are of block-diagonal form, as given in (3.20). ⎞ ⎛ ⎞ ⎛ 1 0 0 −1 0 0 ⎟ ⎜ ⎟ ⎜ σ1 = ⎝ 0 E = ⎝0 1 0⎠ 1 0⎠ 0 0 1 0 0 1 ⎛ ⎜ σ2 = ⎜ ⎝−
1 2 √
−
0 ⎛
− 12
⎜√ 3 C3 = ⎜ ⎝ 2 0
0
⎟ 0⎟ ⎠
− 12
3 2
0 −
√ 3 2
− 12 0
⎛
⎞
√ 3 2
σ3 =
1 2 ⎜√ ⎜ 3 ⎝ 2
1
0 ⎛
⎞
0
− 12
1
3 2
0
⎟ 0⎟ ⎠
− 12 0
⎜ √ 3 C32 = ⎜ ⎝− 2
⎟ 0⎟ ⎠
⎞
√
0
(3.20)
1 √
3 2
− 12 0
⎞ 0 ⎟ 0⎟ ⎠. 1
This is a reducible representation, an important topic to which we will soon return. We have thus far found two irreducible representations for the group C3v . There is a third! We will later prove that there are only three irreducible representations for C3v . The third representation can be determined by considering Figure 3.5. The directed circles located at each corner of the triangle denote a current flow that produces a magnetic dipole. There is a function that describes a magnetic dipole, but here we examine it pictorially. We ask, “What is the effect of acting on the magnetic dipole with the function operators of C3v ?”
y
y
⫻
s^1
x
x ⫻
⫻
Fig. 3.5. The operator σ "1 reverses the direction of the magnetic dipole vector. The directed circles indicate current loops associated with the magnetic dipoles. The reversal of the current direction leads to reversing the dipole direction.
3.1 Representations and realizations
61
The magnetic dipole is a vector quantity, usually represented by an arrow, but the vector is an axial vector (properly named an antisymmetric tensor of rank 2), not a polar vector. Typically in this case, each axial vector denoting a magnetic dipole would be pictured by an arrow (an icon) normal to the plane of the triangle and pointing out of the page for the left-hand side of Figure 3.5. Here, each outward axial vector on the left-hand side is denoted by a dot in the center of each directed circle. The arrows are aimed toward the reader! Now, if each magnetic dipole were truly represented by the icon of an arrow, the mirror reflection operation σ1 would leave the arrows unchanged, and the matrix representation for the mirror reflections would simply be the one-dimensional matrices [ 1 ]. This is the danger in putting all one’s faith in the use of icons to represent the actual physics. If one looks at the current flow, it is clear that the mirror reflections reverse the current flow. Thus the arrow (the icon) now points into the page and is denoted by a cross in the center of each directed circle on the right-hand side of Figure 3.5. The arrow is aimed away from the reader. Mathematically, the same result would have been obtained by considering the angular momentum operator mr × dr/dt. We now have a third irreducible representation generated by function operators in which the group operations are represented by the one-dimensional matrices [ 1 ] for the identity " and the two three-fold function rotation operators C "3 , C "32 , and by the one-dimensional E matrices [ −1 ] for the function reflection operators σ "1 , σ "2 , σ "3 . A discussion of realization versus representations The difference between operations in configuration space and function-space is of such importance that it is timely to discuss once more the idea of these operations. We use the equilateral triangle and the permutation of three objects, from Chapter 2, as examples. We may consider the triangle as a “function” and thus the symmetry operators are then function operators. However, (3.7) allows us to execute the symmetry operations in configuration space, and thus define the corresponding realization of the group operators as in (3.20). In a similar manner, we may consider the three objects to be in function-space, while the boxes among which the objects are permuted are thought of as being in configuration space. In this case, we can think of operations in configuration space as permuting the boxes instead of the objects. We refer to the group of configuration-space operators as a realization of an abstract group. It is a realization in the sense that it is always an isomorphism of the original group. In contrast, we use the term representation in the conventional sense, namely, to describe realizations of group operators in function-space. These representations are homomor"3 on pz phic realizations. We have seen this difference in (3.10) for the operation of C orbitals. Now let us consider the effect of the 3 × 3 matrix representation for the group D3 based on function operators. With the group D3 , the triangle is allowed to rotate about the axes denoted by A, B, C in Figure 2.1. The group elements E, C3 , and C32 remain the same, but the reflection elements σ1 , σ2 , and σ3 from C3v are replaced by two-fold rotation axes perpendicular to the main three-fold axis (henceforth we denote such axes by the
Group representations: concepts
62
symbol U ) U1 , U2 , and U3 for the group D3 . Each of the three 2-fold rotations transform z into −z. Thus, the 3 × 3 function-space matrix representation for D3 is ⎞ ⎞ ⎛ ⎛ −1 0 0 1 0 0 ⎟ ⎟ ⎜ ⎜ U1 = ⎝ 0 E = ⎝0 1 0⎠ 1 0⎠ 0 0 −1 0 0 1 ⎛ ⎜ U2 = ⎝−
√
1 2 √
3 2
0 ⎛
−1 ⎜ √23 C3 = ⎝− 2 0
− 23 − 12 0 √
3 2 − 12
0
⎞ 0 ⎟ 0⎠ −1 ⎞ 0 ⎟ 0⎠ 1
⎛ U3 =
1 ⎜ √23 ⎝ 2
3 2 − 12
0
0
⎛
1 − √2
⎜ C32 = ⎝
⎞ 0 ⎟ 0⎠ −1
√
3 2
0
√
− 23 − 21 0
(3.21)
⎞ 0 ⎟ 0⎠ 1
which is a reducible representation consisting of one irreducible representation identical to that of (3.19) and a second representation that consists of +1 for each of the function " C "3 , and C "32 and −1 for each of the three 2-fold function-rotation operators. operators E, Of course, D3 also has the one-dimensional identity representation consisting of all ones. Thus the point-groups C3v and D3 are each represented by the same three irreducible representations, but with some slight differences in terms of basis functions for the onedimensional representation consisting of three positive unit matrices and three negative unit matrices. An important invariant of a matrix representation of a group is the character of the group elements. The character of a matrix is its trace, that is, the sum of the diagonal elements of the matrix. It is invariant under any similarity transformation. Many applications of group theory in quantum mechanical problems, such as selection rules for electronic and vibrational transitions, can be determined from the use of characters alone. Two representations related by a similarity transformation are called equivalent representations of a group G, namely, " ∼ Γ R " , Γ R (3.22) if,
" = S Γ R " S−1 , Γ R
∀ R ∈ G.
(3.23)
Equivalent representations have the same set of characters, since the trace of a matrix is invariant under a similarity transformation.
3.2 Generation of representations on a set of basis functions Thus far, we have considered in detail only functions that describe atomic p-orbitals. Since the radial parts of these functions are spherically symmetric, only the x, y, or z components are important. This can lead to some confusion between the effects of symmetry operations on coordinates {x, y, z} and operations on functions {x, y, z} that are of the same form as the coordinates.
3.2 Generation of representations on a set of basis functions
63
We now examine operations on higher-order functions to reveal more generally the process of determining the action of a function operator on a function. Before doing so, we first recapitulate what we now know about operations in function-space and their representations. " but we represent A recapitulation We denote a general function-space operator by R, " There are generally at least it with respect to a set of basis vectors by a matrix Γ(R). two or three nonequivalent irreducible representations (Irreps). Thus here we introduce an index µ to label the different Irreps. (µ) " ,R " ∈ G, for each representation, satisfies the group mulEach set of matrices Γ R tiplication table. " operating on a function, we must use (3.6): To determine the effect of R " f (r) = f (R−1 r). R " operating on a basis function can Once a matrix representation is known, the effect of R be expressed as " " fi = fj (µ) Γji (R). R j
Now consider the pair of functions {x2 , y 2 } and ask, “What happens when we operate on these functions with the function operators of C3v ?” "3 . Rewriting (3.8), we have for this case Suppose we wish to find C "3 F (r) = F (C3−1 r), C where
f1 (r) = x2 F (r) = f2 (r) = y 2
(3.24)
(3.25)
"3 F (r) is the new function we wish to determine. and C "3 F (r) from F (r), replace the components of r in F (r) The prescription To generate C by the corresponding components of C3−1 r. We have to generalize our notion of components a bit, as we will see, but for the moment let us proceed as before, finding the coordinates √ ⎡ ⎤ ⎡ 1 ⎤⎡ ⎤ x − 2 − 23 0 x ⎢ ⎥ ⎢√ ⎥⎢ ⎥ −1 ⎢ 3 ⎢ ⎥ ⎥ r =⎢ (3.26) − 12 0⎥ ⎣y ⎦ = C3 r = ⎣ 2 ⎦ ⎣ y ⎦, z z 0 0 1 and, this time, the squares of the coordinates √ 1 2 3 2 3 x + y + xy, 4 4 2 √ 3 1 3 = x2 + y 2 − xy. 4 4 2
x2 = y 2
(3.27)
64
Group representations: concepts
It is clear that the two functions {x2 , y 2 } do not span the entire linear function-space of the C3v operators. That is, the new functions generally cannot be expressed just as linear combinations of the original functions x2 and y 2 . The appearance of a term in xy in the equations for x2 and y 2 makes it clear that in order to span the entire space it is necessary to include xy as a partner basis function. Thus the complete set of basis functions is {x2 , y 2 , xy}. We can incorporate this set of basis functions into a function F(r) defined as ⎞ ⎡ ⎤ ⎛ x2 f1 (r) = x2 ⎢ 2⎥ ⎜ 2⎟ (3.28) F (r) = ⎝ f2 (r) = y ⎠ = ⎣ y ⎦. f3 (r) = xy xy "3 on F(r) will produce a new function. From (3.27) we know the The operation of C relationship between {x2 , y 2 } and {x2 , y 2 }. The other relationship we need is obtained from (3.26): √ √ 1 1 3 3 x y = − x− y x− y 2 2 2 2 √ √ 3 2 3 2 1 =− (3.29) x + y − xy. 2 4 4 We now generalize our understanding of “The prescription” to interpret the components of r to include the basis functions of F (r). " to generate a new set of basis A generalized prescription To use the operator R functions from an old set of basis functions, replace the variables {x, y, z} in the old set of basis functions by the corresponding new variables as given by the components of R−1 r. Thus the basis functions of F (r) are replaced by the basis functions of (3.27) and (3.29), and the new basis functions expressed in terms of the old are given by ⎡ 2 ⎤ x ⎢ 2 ⎥ " ⎢ C3 F (r) = ⎣ y ⎥ ⎦ xy √ ⎤⎡ ⎡ 1 ⎤ 3 3 x2 4 4 2 √ ⎥⎢ ⎢ ⎥ 1 3 ⎢ 2⎥ =⎢ (3.30) − 23 ⎥ ⎦ ⎣ y ⎦. ⎣ 4 4 √ √ 3 3 1 xy − 4 −2 4 "3 on the basis set {x2 , y 2 , xy}. However, This determines the effect of the operation of C " the representative of the function operator C3 is the inverse of the 3 × 3 matrix of (3.30). We must use the inverse matrix, not the transpose, because the matrices here are not unitary matrices. Repeating this procedure with the remaining elements of the group we obtain the set of three-dimensional matrices of C3v that represent the function operators relative to the basis set {x2 , y 2 , xy}. In carrying through this procedure we note that reflection operator
3.2 Generation of representations on a set of basis functions "3 is the inverse of C "32 . Thus, representatives are their own inverses and that C ⎞ ⎞ ⎛ ⎛ √ 1/4 3/4 1 0 0 3/2 √ ⎟ ⎟ (red) (red) "32 ) = ⎜ " =⎜ Γ(C Γ(E) 3/4 1/4 − 3/2⎠ ⎝ √ ⎝0 1 0⎠ √ 0 0 1 − 3/4 3/4 −1/2 ⎞ ⎞ ⎛ ⎛ √ 1/4 3/4 − 3/2 1 0 0 √ ⎟ ⎟ ⎜ (red) (red) "3 ) = ⎜ Γ(C Γ(ˆ σ1 ) = ⎝0 1 0⎠ 1/4 3/2 ⎠ ⎝√3/4 √ 0 0 −1 3/4 − 3/4 −1/2 ⎞ ⎛ √ 1/4 3/4 − 3/2 √ ⎟ ⎜ (red) Γ(ˆ σ2 ) = ⎝ 3/4 1/4 3/2 ⎠ √ √ − 3/4 3/4 1/2 ⎞ ⎛ √ 1/4 3/4 3/2 √ ⎟ ⎜ (red) Γ(ˆ σ3 ) = ⎝ 3/4 1/4 − 3/2⎠ √ √ 3/4 − 3/4 1/2
65
(3.31)
This set of matrices deals with the problem of properly representing the group, as discussed previously with regard to (3.16)–(3.18). However, it is not a group of unitary matrices, i.e. a unitary representation. Let us ask if it is possible to find a unitary representation. This question is raised by the thought that a different set of basis functions might lead to a unitary matrix representation. The answer is, yes, we can choose linear combinations {x2 + y 2 , x2 − y 2 , 2xy} as basis functions, leading to the function ⎛ ⎞ ⎡ ⎤ f1 (r) = x2 + y 2 x2 + y 2 ⎜ ⎟ ⎢ ⎥ F(r) = ⎝f2 (r) = x2 − y 2 ⎠ = ⎣x2 − y 2 ⎦. (3.32) f3 (r) = 2xy 2xy It is necessary to choose 2xy as a basis function, rather than xy, to obtain a unitary representation. (See Exercise 3.5.) Now we know from (3.27) and (3.29) that x2 + y 2 = x2 + y 2 ,
√ 1 3 x2 − y 2 = − (x2 − y 2 ) + (2xy), 2 2 √ 1 3 2 2x y = − (x − y 2 ) − (2xy). 2 2
Thus,
⎡
⎤ ⎡ x2 + y 2 1 ⎢ 2 ⎥ ⎢ ⎣x − y 2 ⎦ = ⎣0 0 2x y
0 −1/2 √ − 3/2
⎤ ⎤⎡ x2 + y 2 0 √ ⎥ ⎥⎢ 3/2⎦ ⎣x2 − y 2 ⎦. 2xy −1/2
(3.33)
(3.34)
"3 relative to the basis functions {x2 + y 2 , x2 − y 2 , 2xy} and the matrix representative of C is the transpose of the 3 × 3 matrix of (3.32), when written to conform with (3.18).
Group representations: concepts
66
Note that the matrix representative is now in block-diagonal form. Indeed, this new set of basis functions leads to all the matrix representatives of C3v being of the same block-diagonal form. Thus, a set of matrices that satisfies the group multiplication table for C3v , as given in Table 2.3, constitutes a matrix representation of this group relative to the basis set {x2 + y 2 , x2 − y 2 , 2xy}, a set of d-orbital functions, given by: ⎞ ⎞ ⎛ ⎛ 1 0 0 1 0 0 ⎟ ⎟ ⎜ (red) (red) " =⎜ Γ(E) Γ(" σ1 ) = ⎝0 1 0⎠ ⎝0 1 0⎠ 0 0 1 0 0 −1 ⎛
1 ⎜ (red) Γ(" σ2 ) = ⎝0 0 ⎛
1 (red) "3 ) = ⎜ Γ(C ⎝0 0
0 −1/2 √ − 3/2 0 −1/2 √ 3/2
⎞ 0 √ ⎟ − 3/2⎠ 1/2 ⎞ 0 √ ⎟ − 3/2⎠ −1/2
⎛
1 ⎜ (red) Γ(" σ3 ) = ⎝0 0 ⎛
1 (red) "32 ) = ⎜ Γ(C ⎝0 0
0 −1/2 √ 3/2
⎞ 0 √ ⎟ 3/2⎠ 1/2
0 −1/2 √ − 3/2
(3.35)
⎞ 0 √ ⎟ 3/2⎠. −1/2
where we used the notation (red) Γ to describe a reducible representation, as will be discussed below. We notice some similarity between (3.35) and (3.20), but why are they different at all? It is simply a result of the order in which we chose the set of basis functions. If we had chosen them in reverse order, i.e. if we had chosen {2xy, x2 − y 2 , x2 + y 2 } we would have obtained (3.20) rather than (3.33). "3 on a set of basis functions by a row Just as in (3.17), we represent the action of C vector operating on the representative matrix. For example, "3 x2 + y 2 , x2 − y 2 , 2xy C ⎛ ⎞ 1 0 0√ 2 ⎜ ⎟ 1 = x + y 2 , x2 − y 2 , 2xy ⎝0 − (3.36) − 23 ⎠ . √2 3 0 −1/2 2 Such matrices can all be decomposed into two smaller matrices, or, conversely, the 3 × 3 matrix can be considered as the augmentation into block-diagonal form of the two smaller "3 is decomposed into matrices. For example, the matrix representing C √ −1/2 − 3/2 √ (1) and . 3/2 −1/2 We write such a relation as ⎛ 1 0 ⎜ −1/2 ⎝0 √ 0 3/2
⎞ 0 √ −1/2 ⎟ − 3/2⎠ = (1) ⊕ √ 3/2 −1/2
√ − 3/2 , −1/2
(3.37)
which reads that the l.h.s. matrix is the direct sum of the two r.h.s. matrices, namely a 1 × 1 and a 2 × 2 matrix. This process, denoted by the encircled plus sign, shows that a "3 can be constructed from matrix representation (Rep) of the group function operator C
Exercises
67
two smaller matrix Reps. We say the l.h.s. matrix Rep is induced from the two r.h.s matrix Reps. Several questions can be raised at this point: • When will the process of decomposition or reduction cease to be possible? In other words, when do the matrix Reps obtained become irreducible? • How do we systematically find irreducible representations (Irreps)? • Is there a finite number of these Irreps? In the present example we were able to reduce the three-dimensional Rep into one 1D and one 2D Reps. Attempts to reduce the two-dimensional Rep will fail. These two Reps, together with the one 1D Rep generated by D3 on pz are the three Irreps of C3v and its isomorphs. There are no others! A very remarkable result emerges from the above presentation; namely, that functions distinguishable under isotropic symmetry become indistinguishable when the symmetry is reduced to that of the triangle. We find that s-, pz - and dz2 -states have the same transformation properties. They transform according to the irreducible representation with all symmetry elements represented by unity! Furthermore, we find that the pairs {px , py } and {dx2 −y2 , dxy } transform like the vector # $ x y . The functions are thus classified according to a set of well-defined categories of symmetries: the Irreps. Any general function can be decomposed into components that belong to welldefined symmetry classes. In some of what we have done thus far in this chapter, we have used prior knowledge of the theory of representations. This was particularly true in the development leading up to (3.35), where we chose appropriate functions to illustrate some important points. Next, we need to further explore and clarify the fundamental principles and concepts of the theory of group representations, and outline the computational techniques that will simplify their construction. This will be done in the next chapter.
Exercises 3.1 Replace the dots in Figure 3.2 with ones, and fill the blank squares with zeroes; show that the resultant matrices satisfy the group multiplication rules of Table 2.1. 3.2 Consider an equilateral triangle with sides of unit length. The triangle is in the xy-plane with its center of gravity at the origin and the coordinates of its apices being √ √ √ (0, 3/3), (1/2, − 3/6), (−1/2, − 3/6). Show that the first apex is taken into the second apex by a clockwise rotation of 120◦ . Let C3 be the operator that rotates the triangle clockwise by 120◦ . Show that "3 that, operating on the function represented the transpose of C3 is the operator C by the vector directed from the origin to the second apex, generates a new function represented by the vector from the origin to the first apex.
68
Group representations: concepts
3.3 Show that the set of matrices analogous to the one in (3.16) do not satisfy the group multiplication table give in Table 2.3. "3 do not 3.4 Show that the set of function-operator matrices as illustrated by (3.16) for C satisfy the group multiplication table. 3.5 Consider x2 − y 2 and xy as two possible basis functions for the group C3v . Writing x = r cos φ, y = r sin φ, show that one must use 2xy rather than xy as a basis function in order that x2 − y 2 and 2xy have the same normalization and thus lead to a unitary matrix representation of C3v . 3.6 The ammonia molecule, NH3 , belongs to the point-group C3v . Consider three functions {fA , fB , fC } that describe the three valence bonds connecting the N atom "3 on the valence bond functions can be with the three H atoms. The operation of C described by ⎞ ⎛ 0 0 1 ⎜ "3 (fA fB fC ) = (fA fB fC ) ⎝1 0 0⎟ C ⎠. 0 1 0 Find the remaining matrices that provide a matrix representation based on the three valence bond functions. Check that the matrices actually obey the group multiplication table. We assume the original basis set to be normalized as well as being orthogonal. Now consider three new (orthonormal) basis functions (vectors) that are linear combinations of the original set: 1 φ1 = √ (fA + fB + fC ) 3 1 φ2 = √ (fA + fB − 2fC ) 6 1 φ3 = √ (fA − fB ) 2 Construct a matrix S whose columns (corresponding to {φ1 , φ2 , φ3 }) are the coefficients of the original basis functions {fA , fB , fC }. Perform the similarity transforma%S for each matrix representative of C3v based on the original basis set to tion S −1 M find the new transformed representation relative to the transformed basis set. What can be said about the new-found representation?
4 Group representations: formalism and methodology
In Chapter 3, we presented the ideas of partner basis functions, on which the action of group elements engender a matrix representation. In this chapter we develop the machinery necessary for the generation of a matrix representation given knowledge of the operation of a function operator on a basis function. We note that when we operate on a set of basis functions that spans the linear space of the symmetry operators, i.e. a complete set, we generate a matrix Rep of the symmetry group, which we now write as " Ψi = R
" Ψj Γji (R),
(4.1)
j
" is the matrix representative of the operator R " with respect to as given in (3.18). Γ(R) the basis functions {Ψj }. This result is implied by the steps leading up to (3.17), where we saw the need to write the operation of a function operator acting on a set of basis functions as the product of a row matrix operating from the left on the matrix Rep of the operation. We now wish to prove that (3.18) and (4.1) are correct. Following that we will develop the theory of group Reps more fully.
4.1 Matrix representations We associate a matrix Γ(R) with each element R of a group, and the assigned matrices obey the group multiplication law. If each matrix is different, the Rep is said to be true, or faithful. We identified the trivial but important nonfaithful case where all group elements are mapped onto unity as the identity or invariant Irrep. Writing (4.1) as S" Ψi =
" Ψj Γji (S),
j
69
(4.2)
Group representations: formalism and methodology
70
" we get and operating on it with R, " S" Ψi = R
" j ) Γji (S) " (RΨ
j
=
" " Γji (S) Ψk Γkj (R)
j,k
=
"S). " Ψk Γki (R
(4.3)
k
Then, using the completeness of the set {Ψi } over the linear space spanned by the group elements, we can write "S) " = " Γji (S), " Γki (R Γkj (R) (4.4) j
thus simultaneously demonstrating, (i) that (4.3) (and hence (4.1)) is the proper way to describe the action of a function operator on a function and, (ii) that the matrix Reps generated obey the group multiplication table. We now state several important properties of group representations. • Equivalent representations Any Rep of a group can be transformed into an equivalent representation by a similarity transformation, as we demonstrated in Section 3.2: " = U Γ(R) " U −1 , Γ (R)
" ∈ G, ∀R
" We can, therefore where U is a unitary square matrix of the same dimension as Rep Γ(R). find an infinite number of Reps equivalent to a given Rep. • Unitary representations We confine our attention to unitary Reps which are guaranteed by the following theorem: • Theorem
Any Rep of a finite group is equivalent to a unitary Rep.
These Reps, however, may be reducible, as we demonstrated in the examples of Chapter 3. A very important test of whether a Rep is reducible or not is provided by Schur’s lemmas. Schur’s first lemma satisfies the relation
Given two inequivalent Irreps µ and ν, the only matrix M that (µ)
" M =M Γ(R)
(ν)
" Γ(R)
is a null matrix. Schur’s second lemma A matrix that commutes with every member of a set of matrices belonging to an Irrep must be a constant, that is, a number times the unit matrix. A simple way of demonstrating Schur’s second lemma is through the following theorem: Theorem If an operator O commutes with all the elements of a group G, then the eigenspace of O is composed entirely of irreducible spaces of G.
4.1 Matrix representations Proof Given
O, R = 0,
∀ R ∈ G,
71
(4.5)
and O Ψλ = λ Ψλ ,
(4.6)
where Ψλ is an eigenfunction of O with eigenvalue λ, the set {Ψλ } forms a complete basis set of the eigenspace of O. From (4.5) and (4.6) we obtain O R Ψλ = R O Ψλ = λ RΨλ ,
(4.7)
and RΨλ is still an eigenvector of O with eigenvalue λ. Thus, for a degenerate, or an invariant, subspace {Ψλ,i } of O, we generate the Rep ⎡ ⎤ Ψλ,1 ⎢ Ψλ,2 ⎥ (λ) ⎥ " " ⎢ Γ(R), (4.8) R ⎢ . ⎥ = Ψλ,1 Ψλ,2 . . . Ψλ,dλ ⎣ .. ⎦ Ψλ,dλ where dλ is the dimension of the invariant space, while ⎡ (λ) ⎤ ⎡ (λ) ⎤ Ψ1 Ψ1 ⎢Ψ(λ) ⎥ ⎢Ψ(λ) ⎥ ⎢ 2 ⎥ ⎢ 2 ⎥ ⎥ ⎢ ⎥ O ⎢ ⎢ .. ⎥ = λ ⎢ .. ⎥ , ⎣ . ⎦ ⎣ . ⎦ (λ) (λ) Ψdλ Ψdλ
(4.9)
which is true for every invariant eigen-subspace of O. Put differently, Schur’s lemma states that, apart from a constant numerical factor, only the unit matrix commutes with all matrices of an Irrep. Although the statement of this lemma might seem to suggest that we determine an Irrep by the exhaustive demonstration of the noncommutativity of its matrices with the set of all matrices of the same dimension, its real significance lies in just the opposite. A key point If we know that a particular operator or observable commutes with all the symmetry group operators, then its matrix representative must commute with any matrix Rep of the group operators, including the Irreps. Consequently, Schur’s lemma then asserts that such quantities have to be represented by a scalar times a unit matrix! To demonstrate the power of this approach let us explore further the properties of matrix Reps.
4.1.1 Diagonal matrix representatives There is an infinite number of equivalent Reps of a finite group. Any matrix M can be transformed into an equivalent matrix M by a similarity transformation M = U −1 M U. The new set of matrices obtained by applying the same similarity transformation to each group element also obeys the group multiplication table.
Group representations: formalism and methodology
72
A particularly interesting transformation is the one that diagonalizes one or more of a set of matrix representatives. It is always possible to diagonalize a real orthogonal matrix, the result generally being a unitary matrix. Let us consider the matrix representative √ 3/2 −1/2 √ Γ(C3 ) = . (4.10) − 3/2 −1/2 We can diagonalize the matrix as follows. First we must find the eigenvalues λ that satisfy the secular equation, which for a general 2 × 2 matrix can be written M − λ M12 11 (4.11) = 0. M21 M22 − λ The value of the determinant of the secular equation can be expressed by the quadratic equation λ2 − (M11 + M22 )λ + (M11 M22 − M12 M21 ) = 0, which has solutions λ=
(M11 + M22 ) ± 2
&
(M11 + M22 )2 − (M11 M22 − M12 M21 ). 4
For the particular case of the group element C3 the solutions are √ 3 1 = e±i2π/3 , λ=− ±i 2 2 and the matrix representative is √ 3/2 0 −1/2 + i "3 ) = √ Γ(C . 0 −1/2 − i 3/2
(4.12)
(4.13)
(4.14)
The matrix elements in (4.14) are complex roots of unity. Indeed, all orthogonal-matrix Reps can be transformed to unitary diagonal matrices whose diagonal elements are complex roots of unity. In the computer method that we shall describe in Chapter 5, and which is based on Dixon’s method, all the matrices will be diagonalized to yield diagonal elements that are complex primitive roots of unity, for the purpose of determining the characters. Thus the diagonal elements will be of the form e±i(2π/n) . In the particular case of the group element C3 , the matrix is then i(2π/3) 0 "3 ) = e Γ(C . (4.15) 0 e−i(2π/3)
4.1.2 Reducible representations Suppose we have a set of matrices Γ(E), Γ(A), . . . , Γ(F ) that satisfy the multiplication table for our example group G6 as given by Table 2.1. That is, Γ(A) Γ(B) = Γ(D),
⇒
AB = D.
4.1 Matrix representations
73
Now transform each matrix in the Rep by a similarity transformation. Then, if Γ (A) = S Γ(A)S, we have −1
Γ (A) Γ (B) = [S −1 Γ(A) S] [S −1 Γ(B) S] = S −1 Γ(A) Γ(B) S = S −1 Γ(AB) S = Γ (AB).
(4.16)
Thus, the transformed matrices Γ (E), Γ (A), . . . , Γ (F ) also form a Rep. This new Rep is equivalent to the original Rep given by the set of matrices Γ(E), Γ(A), . . . , Γ(F ). The two Reps differ only in that they are stated with respect to different sets of coordinate axes or different sets of basis functions. Suppose it is possible to find a similarity transformation that, at once, transforms all the matrices Γ(E), Γ(A), Γ(B), . . . into the block-diagonalized form ⎞ ⎛ Γ1 (A) 0 0 ⎟ ⎜ Γ (A) = S −1 Γ(A) S = ⎝ 0 (4.17) Γ2 (A) 0 ⎠, 0 0 Γ3 (A) where Γ1 (A) is a square matrix with the same dimension as Γ1 (B), . . . , Γ1 (F ). Then, for example, since AB = D in our six-element example group, G6 , we would have Γ1 (A) Γ1 (B) = Γ1 (D), Γ2 (A) Γ2 (B) = Γ2 (D), Γ3 (A) Γ3 (B) = Γ3 (D). That is, the matrix Rep Γ(E), Γ(A), . . . , Γ(F ), is said to be reducible and to have been reduced by the similarity transformation matrix S. In this case, the Rep has been reduced to three different Reps. If there is no similarity transformation that will further reduce all the matrices of a given Rep, the Rep is said to be irreducible, i.e. an Irrep. In many applications, each Irrep has the same transformation properties as a set of degenerate eigenfunctions. We will see examples of this in later sections. There is a particularly interesting set of matrices that forms a reducible Rep of a group, a set that can be block-diagonalized to give all its Irreps. It is known as the regular representation.
4.1.3 The regular representation The regular representation is a useful but artificial construct that is generated on a basis set defined by a faithful mapping of the group elements themselves, i.e. for every group " there corresponds a basis vector R. operator R It can be defined most easily with the help of the group multiplication table. First, construct the multiplication table such that the diagonal elements are just the identity, hence: the first row is labeled by the inverse of the element labeling the first column, the second row is labeled by the inverse of the element labeling the second column, and so forth, as illustrated in Table 4.1 for the group C3v .
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Group representations: formalism and methodology
The regular Rep is probably the most interesting and useful of the reducible Reps, as will become apparent to some extent in this chapter but especially in Chapters 6 and 7, where it is incorporated into a systematic method for constructing the Irreps. The matrices of the regular Rep are constructed with the aid of the following relation: (reg)
"i ) = Γjk (R
1,
"i ; "j R "−1 = R if R k
0,
otherwise.
(4.18)
Example 4.1 The group C 3v Using the regularized multiplication table, Table 4.1, it is straightforward to construct the regular Rep for C3v , using the prescription described by (4.18). To find the regular Rep for σ "1 , for example, we simply substitute a 1 for σ "1 wherever σ "1 appears in Table 4.1. Otherwise the matrix element is 0. This recipe gives the following regular Rep for C3v , Table 4.1. The “regular” multiplication table for C3v .
" E σ "1 σ "2 σ "3 "3 C "32 C
⎛ 1 ⎜0 ⎜ ⎜0 (reg) " =⎜ Γ(E) ⎜ ⎜0 ⎜ ⎝0 0 ⎛
0 ⎜0 ⎜ ⎜1 ⎜ (reg) Γ(" σ2 ) = ⎜ ⎜0 ⎜ ⎝0 0
" E
σ "1
σ "2
σ "3
"32 C
"3 C
" E σ "1 σ "2 σ "3 "3 C "32 C
σ "1 " E "32 C "3 C
σ "2 "3 C " E "32 C
σ "3 "32 C "3 C " E
σ "3 σ2
σ "1 σ "3
σ "2 σ "1
"32 C σ "3 σ "1 σ "2 " E "3 C
"3 C σ "2 σ "3 σ "1 "32 C " E
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
⎞ 0 0⎟ ⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ 1
0 0 0 0 0 1
1 0 0 0 0 0
0 0 0 0 1 0
0 0 0 1 0 0
⎞ 0 1⎟ ⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0
⎛
0 ⎜1 ⎜ ⎜0 ⎜ (reg) Γ(" σ1 ) = ⎜ ⎜0 ⎜ ⎝0 0 ⎛
0 ⎜0 ⎜ ⎜0 ⎜ (reg) Γ(" σ3 ) = ⎜ ⎜1 ⎜ ⎝0 0
1 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 1 0 0 0
⎞ 0 0⎟ ⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟ 0⎠ 0
0 0 0 0 1 0
0 0 0 0 0 1
1 0 0 0 0 0
0 1 0 0 0 0
⎞ 0 0⎟ ⎟ ⎟ 1⎟ ⎟ 0⎟ ⎟ 0⎠ 0
4.1 Matrix representations ⎛
0 ⎜0 ⎜ ⎜0 (reg) "3 ) = ⎜ Γ(C ⎜ ⎜0 ⎜ ⎝1 0
0 0 0 1 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 0 0 1
⎞ 1 0⎟ ⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0
75 ⎛
0 ⎜0 ⎜ ⎜0 (reg) "32 ) = ⎜ Γ(C ⎜ ⎜0 ⎜ ⎝0 1
0 0 1 0 0 0
0 0 0 1 0 0
0 1 0 0 0 0
1 0 0 0 0 0
⎞ 0 0⎟ ⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 1⎠ 0
Generating the regular representation with Mathematica We represent the matrices of the regular Rep as (reg)
"i ) ≡ R[i, j, k], Γjk (R
and recast the condition in (3.18) as Rj = Ri Rk . We can then write the program for generating the regular Rep as:
Do R[i,j,k]=0,{i,1,g},{j,1,g},{k,1,g}
Do
Do
Switch j==m[[i,k]],True,R[i,j,k]=1 ,
{j,1,g},{k,1,g} ,{i,1,g}
Regrep=Array[R,{g,g,g}] Do[Print[MatrixForm[Regrep[[i]]]],{i,1,g}]. The array m[[j,k]] is the multiplication table generated in Chapter 2.
4.1.4 The great orthogonality theorem The great orthogonality theorem is central to the quantitative development of a group theoretical treatment of physical systems. The theorem asserts that Irreps of a group G satisfy the following orthogonality relation:
R∈G
(α)
g " [(α ) Γ(R)] " −1 Γij (R) δαα δii δjj , j i = dα
(4.19)
Group representations: formalism and methodology
76
where dα is the dimension of Irrep (α) Γ. The standard proof of the orthogonality theorem can be found in many books on group theory, such as [1], or in Wigner’s classic book [2]. A more intuitive argument begins with the simple requirement of a group that every element has an inverse: "R "−1 = E. " R Summing over all g elements of the group gives " "R "−1 = g E. R
(4.20)
R
The corresponding equation for the matrix representatives is " [Γ(R "−1 )] = " [Γ(R)] " −1 = g Γ(E), " [Γ(R)] [Γ(R)] R
(4.21)
R
where the second step relies on the relationship " −1 , "−1 )] = [Γ(R)] [Γ(R which follows directly from a comparison of the two equations: " "−1 )] = Γ(E), [Γ(R)][Γ( R " " −1 = Γ(E). [Γ(R)][Γ( R)] Equation (4.21) is a sum rule! Though trivial, it serves as a starting point for the great orthogonality theorem. A convenient device for obtaining orthogonality relations is to insert a unit matrix into (4.21). We can use the identity matrix representative for convenience. It has the correct dimension. Thus " Γ(E) " Γ(R "−1 ) = g Γ(E). " Γ(R) (4.22) R
In terms of a summation over matrix elements, (4.22) becomes " Γjj (E) " Γj i (R "−1 ) = g Γii (E). " Γij (R) jj
(4.23)
R
Now, expressing the identity representative as a Kronecker delta, we have " δjj Γj i (R "−1 ) = g δii . Γij (R) jj
(4.24)
R
The summation on the left-hand side is zero unless j = j . Thus, (α) (α) " (α) Γj i (R "−1 ) = dα " (α) Γj i (R "−1 ) Γij (R) Γij (R) j
R
R
=g δ
ii
δjj ,
(4.25)
4.1 Matrix representations
77
" of dimension dα . Thus, the where now we specify that this applies to each Irrep (α) Γij (R) sum rule for matrix elements leads to the orthogonality relationship (α) " (α) Γj i (R "−1 ) = g δii δjj . Γij (R) (4.26) dα R
Equation (4.26) is a subset of the orthogonality conditions described by (4.22). It also contains a normalization factor 1/dα to account for the dimensionality of the Irrep, which arises from the sum over matrix elements in (4.25). It is important to recognize that the second term of (4.25) is a matrix element of the inverse group operator, which may not belong to the same class as the first operator. This will be important when we later use modular arithmetic to determine group characters and Reps. For a unitary group Rep,
(α)
" Γij (R)
(α )
" ∗= Γi j (R)
R
g δαα δii δjj . dα
(4.27)
This rather imposing relationship is conveniently interpreted as describing the orthonormality of vectors in a g-dimensional space whose axes are labeled by the g elements of the group, and for which the components of a vector are given by a set of matrix elements, all belonging to the same row and column in the matrix Rep. Using our example group G6 , which is of order g = 6, this means that the six matrix elements (α) Γij (E), (α) Γij (A), . . . , (α) Γij (F ), corresponding to the same row and column of each matrix, can be regarded as the six components of a six-dimensional vector vα ij , which is orthogonal to any of the other vectors obtained by a different choice of subscripts, i or j , corresponding to a different α row or column in the irreducible matrix Rep. The vector vij is also orthogonal to any vector designated by a different superscript, α , thus belonging to a different Irrep. How many orthogonal vectors are there? Within a given Irrep (α) Γ of dimension dα , there are d2α matrix elements for each operation of the group. Summing over all nonequiv alent Irreps, there are α d2α vectors; but, in a g-dimensional space there can be only g orthogonal vectors. Thus, we get Burnside’s theorem: α
d2α = g.
(4.28)
A rigorous proof of (4.28), Burnside’s theorem, together with a particularly useful interpretation, is given in Section 4.3. In the case of our six-element example group, we have three Irreps, thus we must have 2 dα = 12 + 12 + 22 = 6 α
as the only possibility. These Irreps were discussed extensively in Chapter 3.
Group representations: formalism and methodology
78
4.2 Character of a matrix representation The character of a matrix Rep of a group G is a set of g numbers χ(E), χ(A), χ(B), . . . , one for each group element, given by
d
χ(R) = Tr Γ(R) = Γii (R),
(4.29)
i=1
where Tr is called the trace, and d is the dimension of Γ. Since a unitary transformation maps a matrix Rep into an equivalent Rep, it preserves the traces of the constituent matrices. Alternatively, we may say that equivalent Reps have identical characters for the matrices representing the same operator. We can use this criterion to differentiate between distinct or equivalent Reps. We also recognize that since class elements are related by conjugacy operations, which in a matrix Rep become similarity transformations, group elements belonging to the same class have identical characters. The identity matrix has only ones along the diagonal, so the trace, or the character, is just the dimension of the Rep. Thus
" = d, χ(E)
(4.30)
where d is the dimension of the Rep. Because the diagonal elements of a diagonalized unitary matrix Rep are roots of unity, d is an upper bound on the character. In the following sections, we will develop some orthogonality relations and equalities that can be used to determine Irrep characters.
4.2.1 Character orthogonality relations We can derive an orthonormality condition for the characters of Irreps by making use of the great orthogonality theorem. When we set j = i, j = i in (4.19) and sum over i, i we obtain, using (4.29), i,i
(α)
Γii (R)
(α )
Γ(R−1 )i i =
R
=
R
i,i
(α)
(α)
Γii (R)
χ(R)
(α )
(α )
Γ(R−1 )i i
χ(R−1 )
R
=
g δαα δii dα i,i
= g δαα .
(4.31)
4.2 Character of a matrix representation
79
Equating the third and fifth terms of these equations gives
(α)
χ(R)
(α )
χ(R−1 ) = g δαα .
(4.32)
R
Thus, the characters are also the components of an orthogonal set of vectors in the g-dimensional group-element space. If we now collect the group elements according to classes, within which the characters are identical, we can change the last equation to a sum over classes, yielding
nc(i)
(α)
χ(Ci )
(α )
χ(C−1 i ) = g δαα ,
(4.33)
Ci
where χ(Ci ) is the character of class Ci , and χ(C−1 i ) is that of its inverse class, and nc(i) is the number of elements in Ci . The summation is over all classes in the group. This orthogonality relationship can be interpreted as follows: The characters of the various Irreps form an orthogonal set of vectors in a vector space where axes are labeled by the " A second character orthogonality relation is classes Ci rather than the group elements R. given by α
(α)
χ(Ci )
(α)
χ(C−1 j ) =
g δij . nc(i)
(4.34)
The proof is left as Exercise 4.3. Suppose there are l classes in the group. Then the space in which axes are labeled by classes is an l-dimensional space. Since there can be l orthogonal vectors in an l-dimensional space, and the orthogonality is between characters from different Irreps, there can be a maximum of l Irreps. Thus l, the number of classes, is an upper bound on the number of Irreps. Thus we have the important equality:
Number of Irreps = Number of classes.
(4.35)
We are now in a position to determine the class characters for the group C3v . Of course, we already know the matrix Irreps for the group C3v and so we could calculate the characters directly by summing the diagonal elements of the representative matrices. However,
80
Group representations: formalism and methodology
we want to show that it is possible to determine the characters without explicit knowledge of the matrix Irreps.
Example 4.2 The group C3v consists of three classes. Thus, from (4.35) we know there must be three Irreps. Then, from (3.28) and the discussion thereafter we know that these Irreps must be of dimensions 1, 1, and 2. We have denoted them earlier by (1) Γ, (2) Γ, and (3) Γ, respectively. One of these Irreps must be the one-dimensional invariant Irrep, which we take to be (1) Γ. Thus, the first row of the character table, corresponding to (1) Γ, consists of all ones. The first column of the table, corresponding to class C1 = E must be equal to the dimension of the Irrep, as we know from (4.25). Thus we can immediately write down a partial character table, as follows. Table 4.2. Partial character table for the point-group C3v .
Γ1 Γ2 Γ3
χ(C1 ) χ(C−1 1 )
χ(C2 ) χ(C−1 2 )
χ(C3 ) χ(C−1 3 )
1 1 2
1
1
Notice that the classes of C3v are such that C−1 = Ci . i We can fill in the other values of the table by trial and error, making use of (4.33), which is an orthogonality relationship between rows of the character table. If we take class C2 to be the class consisting of the two elements {C3 , C32 }, and C3 to consist of {σ1 , σ2 , σ3 }, then the orthogonality relationship for characters is satisfied only by Table 4.3. Columns of the table are orthogonal, as expected from (4.34). Table 4.3. Character table for the point-group C3v .
Γ1 Γ2 Γ3
χ(C1 )
χ(C2 )
χ(C3 )
χ(C−1 1 )
χ(C−1 2 )
χ(C−1 3 )
1 1 2
1 1 −1
1 −1 0
The example we have just gone through typifies what is generally done to determine character tables, but it presents difficulties in practice for large tables. There is
4.2 Character of a matrix representation
81
a systematic way of determining characters, known as Burnside’s method; we present it in Section 4.3.
4.2.2 Character decomposition " is the character of a reducible Rep of element R " ∈ G, then it is possible to If χ(R) decompose it into Irrep characters according to (α) (red) " = " χ(R) χ(R), (4.36) red α (red)
α
where red α is the number of times the αth Irrep appears in the reducible Rep (red) Γ; it is called the multiplicity or the decomposition frequency of Irrep α in the reducible Rep. We can therefore write (red) Γ in the form of a direct sum over Irreps as
(red)
Γ=
'
red α
(α)
Γ.
(4.37)
α
we determine red α from the following relation:
1 red α = g R
(α)
χ(R−1 )
(red)
χ(R),
(4.38)
or,
Nc
1 nc(i) red α = g i=1
(α)
χ(C−1 i )
(red)
χ(Ci ),
(4.39)
where Nc is the number of classes and nc(i) is the number of elements in class Ci . A proof of (4.38) and (4.39) is easily obtained using (4.32) or (4.33). Equations (4.37)–(4.39) will be used extensively in subsequent chapters. Let us now use (4.38) or (4.39) to determine the character decomposition frequencies for the regular Rep. We know from the definition of the regular Rep as given by (4.18), and from the example of (4.19), that the identity element only has nonzero character. Thus, we find that (reg)
"i ) = χ(R
g,
0,
"i = E; " if R otherwise.
(4.40)
Group representations: formalism and methodology
82
Now, using this result in (4.39) we get NC
1 (reg) reg α = nc(i) (α) χ(C−1 χ(Ci ) i ) g i=1
1 (α) χ(E −1 ) (reg) χ(E) g 1 = × dα × g = dα , g =
(4.41)
This is a very important result. It is a restatement of Burnside’s theorem as given in (4.35), but now given in a way that is particularly insightful and useful. It is simply this: Burnside’s theorem
An Irrep of dimension dα appears dα times in the regular Rep.
The character turns out to be the single most useful invariant of a Rep. This is because the sum of the eigenvalues is given by the sum of the diagonal elements of a matrix in any Rep, and it is invariant under a similarity transformation.
4.2.3 Class matrices and Dirac characters In a given Irrep
(α)
Γ we define a class matrix as
(α)
Ci
=
(α)
" . Γ(R)
(4.42)
R∈C i
The class matrices are constructed by adding corresponding matrix elements (same row and column) from each matrix for each group element in the class, as illustrated in the following examples.
Example 4.3 The class matrices for the two-dimensional Irrep 1 0 (3) C1 = (3) Γ(E) = 0 1 (3)
Γ of C3v are
Γ(C32 ) √ 3/2 −1/2 −1/2 √ = + √ − 3/2 −1/2 3/2 −1 0 = 0 −1
C2 =
(3)
(3)
Γ(C3 ) +
(3)
√ − 3/2 −1/2
4.2 Character of a matrix representation (3)
C3 =(3) Γ(σ1 ) + (3) Γ(σ2 ) + (3) Γ(σ3 ) √ −1 0 1/2 − 3/2 1/2 √ = + + √ − 3/2 0 1 −1/2 3/2 0 0 = 0 0
83
3/2 −1/2
√
The class matrix for the ith class in the regular Rep can be written as (reg)
Ci =
(reg)
Γ(i, j),
(4.43)
j
where (reg) Γ(i, j) is the regular Rep of the jth element in class Ci . The regular class matrix for the identity class is the unit matrix of dimension g.
Example 4.4 For the regular class matrices of the group C3v , other than the identity matrix, we get (reg)
"3 ) + (reg) Γ(C "32 ) C2 =(reg) Γ(C ⎞ ⎛ 0 0 0 0 0 1 ⎜0 0 1 0 0 0⎟ ⎟ ⎜ ⎜0 0 0 1 0 0⎟ ⎟ ⎜ =⎜ ⎟ + ⎜0 1 0 0 0 0⎟ ⎟ ⎜ ⎝1 0 0 0 0 0⎠ 0 0 0 0 1 0 ⎞ ⎛ 0 0 0 0 1 1 ⎜0 0 1 1 0 0⎟ ⎟ ⎜ ⎜0 1 0 1 0 0⎟ ⎟ ⎜ =⎜ ⎟ ⎜0 1 1 0 0 0⎟ ⎟ ⎜ ⎝1 0 0 0 0 1⎠ 1 0 0 0 1 0
(reg)
C3 =(reg) Γ(" σ1 ) + ⎛
0 ⎜1 ⎜ ⎜0 ⎜ =⎜ ⎜0 ⎜ ⎝0 0
1 0 0 0 0 0
0 0 0 0 1 0
(reg)
0 0 0 0 0 1
Γ(" σ2 ) +
0 0 1 0 0 0
⎛
0 ⎜0 ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎜ ⎝0 1
(reg)
⎞ ⎛ 0 0 ⎜0 0⎟ ⎟ ⎜ ⎟ ⎜1 0⎟ ⎜ ⎟+ ⎜ ⎜0 1⎟ ⎟ ⎜ ⎝0 0⎠ 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 1 0 0 0 0
1 0 0 0 0 0
⎞ 0 0⎟ ⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 1⎠ 0
1 0 0 0 0 0
0 0 0 0 1 0
0 0 0 1 0 0
⎞ 0 1⎟ ⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0
Γ(" σ3 ) 0 0 0 0 0 1
84
Group representations: formalism and methodology ⎛ 0 ⎜0 ⎜ ⎜0 ⎜ + ⎜ ⎜1 ⎜ ⎝0 0 ⎛
0 ⎜1 ⎜ ⎜1 ⎜ = ⎜ ⎜1 ⎜ ⎝0 0
0 0 0 0 1 0 1 0 0 0 1 1
0 0 0 0 0 1 1 0 0 0 1 1
1 0 0 0 0 0 1 0 0 0 1 1
⎞ 0 0⎟ ⎟ ⎟ 1⎟ ⎟ 0⎟ ⎟ 0⎠ 0
0 1 0 0 0 0
⎞ 0 1⎟ ⎟ ⎟ 1⎟ ⎟ 1⎟ ⎟ 0⎠ 0
0 1 1 1 0 0
Class matrices also satisfy the class multiplication relations as defined in Section 2.2: Ci Cj =
NC
hijk Ck ,
(4.44)
k=1
namely,
(α)
Ci
(α)
Cj =
NC
hijk
(α)
Ck .
(4.45)
k=1
However, unlike the multiplication of classes in (4.44), where the coefficients are uniquely determined by the multiplication process, this is not so for (4.45). That is, the actual multiplication of two class matrices gives a single matrix which can be decomposed according to (4.45), but the coefficients must be known from (4.44).
Example 4.5 Consider multiplication of the class matrix (3) C2 by itself: −1 0 −1 0 (3) (3) C2 C2 = 0 −1 0 −1 1 0 = . 0 1 Comparison with the results of Example 4.3 might (incorrectly) suggest the decomposition is given by (3)
(3)
C2 C2 =
(3)
C1
WRONG!
4.3 Burnside’s method
85
with h221 = 1 and all other hijk = 0. However, by analogy with Example 2.7, we know that C2 C2 = 2C1 + C2 , where the superscripts (3) have been deleted to clearly emphasize that h221 = 2 and h222 = 1. The point is that one cannot uniquely decompose the product of two class matrices from a knowledge of the product matrix alone. It requires that the coefficients hijk be known from (4.44).
The trace of a class matrix has the form " = nc(i) χ(Ci ), Tr (α) Γ(R) Tr (α) Ci =
(4.46)
R∈C i
where nc(i) is the number of elements in class Ci . A class sum commutes with all the elements of the group (2.17), thus Schur’s lemma requires that its matrix representative in (α) Γ be of the form
(α)
Ci =(α) λi I,
(4.47)
where the scalar (α) λi is a class constant or Dirac character [14], and I is the unit matrix of dimension dα . Consequently, Tr(Ci ) =(α) λi dα .
(4.48)
The Dirac character is an eigenvalue solution to an equation used to determine the group characters by Burnside’s method, which we develop in the next section. Combining (4.46) and (4.48) we obtain the following relationship between the character and the Dirac character (class constant) of a class:
(α)
(α)
λi =
nc(i)χi dα
.
(4.49)
In the next section we learn how to determine the Dirac character of all the classes and all the Irreps in one fell swoop by means of Burnside’s method. Equation (4.49) then allows us to convert to the character χi that is used in actual group theory calculations.
4.3 Burnside’s method We learned in Section 4.2 how to construct character tables for Irreps by using the simple rules of orthogonality relations for characters; sometimes this requires some guessing and checking. There are other more systematic methods that can be used, generally involving the solution of sets of equations. A particularly useful and informative method is that
86
Group representations: formalism and methodology
of Burnside [3]. It is essentially a transformation of the problem to the solution of an eigenvalue equation. Burnside’s method for the calculation of characters of Irreps is based on the diagonalization of the class constant matrices. It is systematic and unambiguous, but it may suffer from numerical round-off errors that result from computer solutions of eigenvalue problems, especially for large groups. Dixon [4] proposed a method for obtaining the eigenvalues and eigenvectors, which we will describe in Chapter 5, that avoids the numerical errors inherent in the traditional Burnside’s approach. This method employs a clever transformation of the problem to one in “prime number” space, where the solutions are integers and hence can be found exactly and then transformed back to the space of real numbers. Thus, for both tutorial and historical reasons, a treatment of Burnside’s method provides an excellent introduction to Dixon’s method. We start with (4.45), the class multiplication relation for class sum matrices, (α)
Ci
(α)
Cj =
NC
(α)
hijk
Ck ,
k=1
and substitute for the Ci s from (4.47). We obtain the following relations among the class constants: (α)
λi
(α)
λj =
hijk
(α)
λk .
(4.50)
k
The coefficients hijk are independent of the Irrep; they are properties of the group. They can be considered as the elements of a class-coefficient matrix H (i) , with rows and columns labeled by j and k, namely ⎞ ⎛ hi11 hi12 hi13 · · · ⎟ ⎜h ⎜ i21 hi22 hi23 · · ·⎟ H (i) = ⎜ ⎟, · · · · ·⎠ ⎝ · · · · ··· and the class constants
(α)
λi can be considered as components of a vector ⎞ ⎛ (α) λ1 ⎜(α) ⎟ (α) λ2 ⎟ Λ=⎜ ⎝ . ⎠. ..
For each class i, the set of equations, given by (4.50) when j assumes all possible values, can thus be written as an eigenvalue equation, namely, H (i)
(α)
Λ =(α) λi
(α)
Λ.
(4.51)
Notice that the eigenvalues, which are the class constants (α) λi , also make up the components of the eigenvectors. Hence, the class matrices H (i) share the same eigenvectors, and
4.3 Burnside’s method
87
can be simultaneously diagonalized by a similarity transformation, V , comprising these eigenvectors V −1 H (i) V = ∆, to yield the diagonal matrix ∆, whose diagonal elements are the eigenvalues of H (i) . This is a powerful statement. Every class matrix is diagonalized by the same similarity transformation! This is a wonderful byproduct of the special connection between eigenvalues and the components of eigenvectors for this particular circumstance. We demonstrate it in Example 4.6. r There are numerous methods to diagonalize a matrix. However, Mathematica provides three functions for the solution of a matrix eigenvalue problem:
gives the complete set of eigenvalues and corresponding eigenvectors, Eigenvalues[H] gives only the eigenvalues, Eigenvectors[H] gives only the eigenvectors.
Eigensystem[H]
r The following program illustrates the use of these Mathematica functions in the construction of the character table.
h = Array[a,{nc, nc, nc}]; Print["CLASS MULTIPLICATION MATRICES"]; Do[Print["H(", i, ") = ", MatrixForm[h[[i]]]],{i, 2, nc} ]; Do[d[mu] = 1,{mu, 1, nc}]; nv = 1; nm = 1; Λ = {}; While[nv < nc, nm++; Switch[nm≤ nc, True, EN = Eigenvalues[h[[nm]]]; Do[ EG = EN[[iv]]; Switch[Length[Position[EN, EG]] == 1, True, EV = Eigenvectors[h[[nm]] ];
Group representations: formalism and methodology
88
Switch[FreeQ[Λ, EV[[iv]]], True, AppendTo[Λ, EV[[iv]]];nv++] ],{iv, 1, nc} ], False, ncl = 1; While[ncl < nc - 1, ncl++; icl = ncl; While[icl < nc, icl++; Do[bb[i, j] = a[ncl, i, j] + a[icl, i, j], {i,1, nc}, {j, 1, nc} ]; Array[bb,{nc, nc}]; Do[ EG = EN[[iv]]; Switch[Length[Position[EN, EG]] == 1, True, EV = Eigenvectors[hh]; Switch[FreeQ[Λ, EV[[iv]]], True, AppendTo[Λ,EV[[iv]]]; nv++ ], {iv, 1, nc} ] ] ] ] ]; Print["Dirac characters : ", MatrixForm[Λ]]; Do[ din = 0; Do[ din += Λ[[iv, ic]]Conjugate[Λ[[iv, ic]]]/ Length[LC[[ic]]], {ic, 1, nc} ]; d[iv] = Simplify[gq/din]; d[iv] =
(
d[iv];
Print["The Irrep dimensions are"] Print["d[", iv, "] = ", d[iv]], {iv, 1, nc}]; χ = {};
4.3 Burnside’s method
89
Do[ Do[ chi = d[mu] Λ[[mu,ic]]/Length[LC[[ic]]]; AppendTo[χ, chi], {ic,1, nc} ], {mu, 1, nc} ]; χ = Partition[χ, nc]; Print["The Irrep characters are"]; Do[Print["Γ[", mu, "] :", χ[[mu]]], {mu, 1, nc}
Example 4.6 The group C3v contains the classes C1 = E, C2 = C3 , C32 and C3 = σ1 , σ2 , σ3 . We begin by using class multiplication to determine the coefficients hijk that provide the matrix elements for H (i) . With the help of Table 2.1 for group multiplication we find, for example, that C2 C2 = (C3 + C32 )(C3 + C32 ) = C32 + C32 C3 + C3 C32 + C34 = C32 + E + E + C3 = 2C1 + C2 . From this we see that h221 = 2 and h222 = 1. Continuing in this fashion for all other products of classes, we find h111 = h122 = h133 = h212 = h222 = h313 = 1, h221 = h233 = h323 = 2, h331 = h332 = 3. All other hijk = 0. We use the coefficients hijk to construct ⎛ h111 h112 ⎜ H (1) = ⎝h121 h122 h131 h132 ⎛
H (2)
H (3)
h211 ⎜ = ⎝h221 h231 ⎛ h311 ⎜ = ⎝h321 h331
h212 h222 h232 h312 h322 h323
the matrices H (i) : ⎞ ⎞ ⎛ 1 0 0 h113 ⎟ ⎟ ⎜ h123 ⎠ = ⎝0 1 0⎠ , 0 0 1 h133 ⎞ ⎛ 0 h213 ⎟ ⎜ h223 ⎠ = ⎝2 0 h233 ⎞ ⎛ 0 h313 ⎟ ⎜ h323 ⎠ = ⎝0 3 h333
1 1 0 0 0 3
⎞ 0 ⎟ 0⎠ , 2 ⎞ 1 ⎟ 2⎠ . 0
90
Group representations: formalism and methodology
Applying the function Eigenvectors to the matrix H (3) we obtain the set of three class constant eigenvectors: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (1) (2) (3) Λ = ⎝2⎠ , Λ = ⎝ 2⎠ , Λ = ⎝−1⎠ . 3 −3 0 We normalized the eigenvectors such that the class constants corresponding to the identity element are set to unity. This is necessary in order to satisfy the relation between the class constants and the class characters, as stated in (4.49), and ensures that the columns of the matrix V are eigenvectors whose elements are exactly equal to the eigenvalues. Moreover, we notice that because the matrices we wish to diagonalize are not all symmetric, the eigenvectors are not orthogonal, but the “vectors” we eventually determine (whose components are the class characters) will be orthogonal in the sense of (4.32) and (4.34). The class constant eigenvectors then form the similarity transformation matrix ⎞ ⎛ 1 1 1 ⎟ ⎜ V = ⎝2 2 −1⎠ 3 −3 0 with its inverse given by
⎛
V −1
1/6 ⎜ = ⎝1/6 4/6
1/6 1/6 −2/6
⎞ 1/6 ⎟ −1/6⎠ . 0
Now, as an example, we apply this transformation to diagonalize the matrix H (3) in order to find the class constants for the class C3 . Of course, we actually already know them from the matrix V , but it is instructive to carry through with the matrix diagonalization to assure ourselves that everything is consistent. ⎞ ⎞⎛ ⎞⎛ ⎛ 1 1 1 0 0 1 1/6 1/6 1/6 ⎟ ⎟⎜ ⎟⎜ ⎜ V −1 H (3) V = ⎝1/6 2 −1⎠ 1/6 −1/6⎠ ⎝0 0 2⎠ ⎝2 3 −3 0 3 3 0 4/6 −2/6 0 ⎞ ⎛ (1) ⎛ ⎞ λ3 0 0 3 0 0 ⎟ ⎜ ⎜ ⎟ (2) = ⎝0 −3 0⎠ = ⎝ 0 λ3 0 ⎠. (3) 0 0 0 0 0 λ3
Table 4.4. Class constants for the point-group C3v .
(1)
Γ Γ (3) Γ (2)
λ(C1 )
λ(C2 )
λ(C3 )
1 1 1
2 2 −1
3 −3 0
4.3 Burnside’s method
91
We have now obtained the class constants for the class C3 of the point-group C3v by diagonalizing the matrix H (3) . Each of the columns of the matrix V that yields the diagonalization provides the class constants for one of the Irreps. The columns of V become the rows of the class constant table, as in Table 4.4.
A key point The matrix V that diagonalizes the class-coefficient matrix H (i) is of dimension NC × NC , where NC is the number of classes and also equals the number of Irreps. The columns of V are eigenvectors whose components are the class constants for an Irrep. Each column is distinct and each corresponds to a single Irrep. We have demonstrated how to determine class constants for each Irrep in an unambiguous way. We now obtain the corresponding irreducible character table with the aid of (4.49), which can be written as
(α)
χi
=
dα nc(i)
(α)
λi ,
(4.52)
where nc(i) is the number of group elements in class Ci . To calculate the characters from the class constants we need to know the dimensions of the Irreps. We make use of the orthogonality relationship given by (4.33), which, together with (4.52), gives nc(i) (α) χi (α) χ∗i = g. i
From (4.33) and (4.52) we find g
d2α = i
((α) λi
(α) λ∗ /nc(i)) i
.
(4.53)
We can now determine the class characters.
Example 4.7 Using (4.53) we find the dimension d3 of the Irrep d23 =
6 1 1
+
1 2
+
0 3
(3)
Γ for the group C3v :
= 4.
Thus d3 = 2. Similarly, we find d1 and d2 = 1. From Table 4.3 and (4.52), together with our knowledge of the dimensions of the Reps we find, with no trial and error, that we reproduce the character Table 4.3 for C3v .
Group representations: formalism and methodology
92
We notice that the class matrix H (3) has three nondegenerate eigenvalues, and thus the corresponding eigenvectors can be unambiguously identified with the different Irreps. If we had chosen to diagonalize H (2) instead, we would have obtained the eigenvalues {2, 2, −1}. The eigenvectors corresponding to the two-fold degenerate eigenvalue, 2, cannot be unambiguously identified with Irreps, since any linear combination of the eigenvectors [1, 2, 3] and [1, 2, −3] constitutes an eigenvector of H (2) with eigenvalue 2. To resolve this shortcoming, when encountered, we pick only the eigenvector corresponding to the nondegenerate eigenvalue, −1, and seek the remaining two by proceeding to diagonalize H (3) . A very tricky situation is encountered when attempting to use Burnside’s method for the group C4v .
Example 4.8 C4v has five classes: C1 = E, C2 = {C4 , C4−1 }, C3 = {σ1 , σ2 }, C4 = {C2 }, C5 = {σ1d , σ2d }. The corresponding class matrices, apart from the identity matrix, are given by ⎛
0
H
(2)
⎜ ⎜2 ⎜ ⎜ = ⎜0 ⎜ ⎜0 ⎝ 0 ⎛
0
H
(4)
⎜ ⎜0 ⎜ ⎜ = ⎜0 ⎜ ⎜0 ⎝ 2
1
0
0
0
0
2
0
0
0
1
0
0
0
2
0
0
0
0
0
2
0
2
0
0
0
0
0
0
0
2
⎞ 0 ⎟ 0⎟ ⎟ ⎟ 2⎟ ⎟ 0⎟ ⎠ 0 ⎞ 1 ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟ 1⎟ ⎠ 0
⎛
0
H
(3)
⎜ ⎜0 ⎜ ⎜ = ⎜2 ⎜ ⎜0 ⎝ 0 ⎛
0
H
(5)
⎜ ⎜0 ⎜ ⎜ = ⎜0 ⎜ ⎜1 ⎝ 0
0
1
0
0
0
0
0
0
2
1
0
0
0
2
0
0
0
1
1
0
0
0
1
0
0
0
0
0
0
0
⎞ 0 ⎟ 2⎟ ⎟ ⎟ 0⎟ ⎟ 0⎟ ⎠ 0 ⎞ 0 ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟ 0⎟ ⎠ 1
We find that H (2) has eigenvalues {−1, 1, 1, 1, 1}, and H (3) , H (4) , and H (5) have the same eigenvalues {−2, −2, 0, 2, 2}. The nondegenerate eigenvector in each case is the same, namely, [1, 0, 0, −1, 0]. Note that eigenvalues are enclosed in braces {} and eigenvectors are enclosed in brackets [ ]. In order to determine the remaining eigenvectors we use the fact that linear combinations of these matrices will have the same set of eigenvectors, but new eigenvalues. The new nondegenerate eigenvalues may reveal the sought-after eigenvectors. In fact, for the case of C4v the remaining eigenvectors [1, 2, 2, 1, 2], [1, −2, −2, 1, 2], [1, −2, 2, 1, −2], [1, 2, −2, 1, −2] are obtained from pair sums of the class matrices, i.e. H (i) + H (j) .
4.3 Burnside’s method
93
Table 4.5. Class constants for the point-group C4v . λ(C1 )
λ(C2 )
λ(C3 )
λ(C4 )
λ(C5 )
1 1 1 1 1
2 2 −2 −2 0
2 −2 2 −2 0
1 1 1 1 −1
2 −2 −2 2 0
(1)
Γ Γ (3) Γ (4) Γ (5) Γ (2)
We next use (4.53) to obtain the dimensions of the Rep. We find [ 1, 0, 0, −1, 0 ] :
d2 =
[1, −2, −2, 1, 2] :
d2 =
[ 1, 2, 2, 1, 2 ] :
d2 =
[1, −2, 2, 1, −2] :
d2 =
[1, 2, −2, 1, −2] :
d2 =
8 1 1
+
0 2
+
1 1
+
4 2
+
1 1
+
4 2
+
1 1
+
4 2
+
1 1
+
4 2
+
0 2
+
1 1
+
0 2
+
1 1
+
4 2
+
1 1
+
4 2
+
1 1
+
4 2
+
1 1
+
4 2
= 4 → d = 2,
8 4 2
= 1 → d = 1,
8 4 2
= 1 → d = 1,
8 4 2
= 1 → d = 1,
8 4 2
= 1 → d = 1.
Finally, we use (4.52) to obtain the characters, starting with (1)
(1)
(1)
(2)
(2)
(2)
(1)
(1)
[ 1, 2, 2, 1, 2 ] : χ1 = 1, χ2 = 1, χ3 = −1, χ4 = 1, χ5 = 1, (2)
(2)
[1, 2, −2, 1, −2] : χ1 = 1, χ2 = 1, χ3 = 1, χ4 = 1, χ5 = −1, (3)
(3)
(3)
(4)
(4)
(4)
(5)
(5)
(3)
(3)
[ 1, −2, 2, 1, −2] : χ1 = 1, χ2 = −1, χ3 = 1, χ4 = 1, χ5 = −1, (4)
(4)
[ 1, −2, −2, 1, 2] : χ1 = 1, χ2 = −1, χ3 = −1, χ4 = 1, χ5 = 1, (5)
(5)
(5)
[ 1, 0, 0, −1, 0 ] : χ1 = 2, χ2 = 0, χ3 = 0, χ4 = −2, χ5 = 0,
94
Group representations: formalism and methodology Table 4.6. Characters for the point-group C4v .
(1)
Γ Γ (3) Γ (4) Γ (5) Γ (2)
χ(C1 )
χ(C2 )
χ(C3 )
χ(C4 )
χ(C5 )
1 1 1 1 2
1 1 −1 −1 0
1 −1 1 −1 0
1 1 1 1 −2
1 −1 −1 1 0
which are given in Table 4.6 in a compact form.
Exercises 4.1 Show that a similarity transformation relating two equivalent unitary Irreps must be unitary if its determinant is 1, i.e. if it is unimodular. 4.2 Use Schur’s lemma to demonstrate that all the Irreps of an Abelian group are one dimensional. Hence the number of Reps equals the order of the group. 4.3 Show that the character is invariant under a similarity transformation. 4.4 Prove the character orthogonality relationship, (α) g χ(Ci ) (α) χ∗ (Cj ) = δij , nc(j) α for the complete set of unitary Irreps of a group G. This is useful for checking the orthonormality of columns in a character table such as in Table 4.1. (Hint: Use the first orthogonality relation to demonstrate the unitarity of the matrix 1/2 nc(i) (α) Uαi = χ(Ci ), U U ∗ = E, g and hence show that simple commutation of this product yields the second orthogonality relation.) 4.5 Show that (µ) χ(R) = 0, R∈G
4.6
4.7 4.8 4.9
for any Irrep (µ) of G except the identity Irrep. Since the characters form an orthogonal set of vectors, as described by (4.33) and "−1 ), sum over group elements R, " (4.34), multiply (4.37) on both sides by (α ) χ(R collect elements into classes and obtain (4.40). Use Burnside’s method to determine the Irreps and characters of the point-group C3 . Do not use a computer program, rather work it out by hand. Construct the character table for the group C4v following the steps of Example 4.2. Construct the character table for the tetrahedral point-group T .
Computational projects
95
4.10 Construct the class matrices for the two-dimensional Irrep of the group C4v : 1 0 0 −1 0 1 −1 0 −1 E= , C4 = , C4 = , C2 = , 0 1 1 0 −1 0 0 −1 σx =
1 0
0 −1 , σy = −1 0
0 0 , σ1 = 1 1
1 0 , σ2 = 0 −1
−1 . 0
Show that they commute with all the corresponding matrix operators of the group. Hence, according to Schur’s lemma they should have the form of a constant (the Dirac character) times the two-dimensional unit matrix. Diagonalize these class matrices and obtain the corresponding Dirac characters. 4.11 Show that a necessary and sufficient condition for the irreducibility of a Rep (α) of a finite group G is 2 1 (α) χ(R) = 1. g R∈G 4.12 Transform the permutations obtained in Exercise 2.16 for the point-group C4v into matrix form, and show that it forms a matrix Rep of C4v of dimension 4. Show that this Rep is reducible. Determine the multiplicities of the Irreps of C4v in this Rep. 4.13 Determine the multiplicities of the three Irreps of C3v in its regular Rep.
Computational projects (i) Write a program to generate the regular Rep. Check that the matrix representatives for C3v are given correctly by (3.49). (ii) (a) Augment the class multiplication matrices program, developed in Chapter 2, with matrix diagonalization capabilities (either by using diagonalization subroutines, or using Mathematica functions such as Eigenvalues[m], Eigenvectors[m], or Eigensystem[m]). (b) Use this new program to calculate the Dirac characters of the groups: C6v , D3h , Td . (c) Determine the dimensionality of the respective Irreps. (d) Use (4.47) to construct the corresponding irreducible character tables.
5 Dixon’s method for computing group characters
The characters of group elements in a matrix representation are complex numbers. To determine these irreducible characters with Burnside’s method we begin by diagonalizing a matrix of coefficients hijk that describe the results of class multiplication. The eigenvalues obtained in this manner are simply related to the irreducible characters, as described in Chapter 4. If only groups of small order were of interest, Burnside’s method might be sufficient. However, for large groups, the introduction of numerical round-off errors can be serious. Moreover, for many purposes one wants the characters expressed in algebraic form. In 1967, John Dixon [4] devised a powerful method by which the characters can be calculated in algebraic form without suffering from numerical errors. The essence of Dixon’s method is a transformation of the Burnside method from the field of complex numbers to the field of integers modulo p. For a suitable prime number p, the modular integer characters can be easily and exactly determined. They can then be transformed back to the ordinary irreducible characters in the field of complex numbers. With Dixon’s method, one does not actually diagonalize the class-coefficient matrix in order to determine the eigenvalues. Rather, one assumes an eigenvalue on a trial basis and then checks to see if it is correct. This is made possible because the solutions are all √ integers mod p where, as we shall see, p > 2 g. Even for a group with as many as, say, 106 elements, we need only choose p > 2000. It is quite simple and quick to check all of several thousand integers to see which ones satisfy the eigenvalue equations. In this chapter we begin by considering Burnside’s eigenvalue equation transformed to the field of integers modulo p, which we assume to be rigorous. After familiarizing ourselves with its solution, we introduce Dixon’s method for irreducible characters. In Appendix 1, we demonstrate the validity of Burnside’s eigenvalue equation in modular form. 5.1 The eigenvalue equation modulo p Burnside’s eigenvalue equation, which was described in Chapter 4 is Hi
(α)
Λ=
(α)
λi
(α)
Λ.
(5.1)
The equivalent equation in the field of integers modulo p is Hi Θ
(α)
Λ = Θ (α) λi Θ (α) Λ ,
96
(5.2)
5.1 The eigenvalue equation modulo p where the mapping Θ Θ
(α)
(α)
97
Λ is given by
# $ Λ = Θ((α) λ1 ), Θ((α) λ2 ), . . . , Θ((α) λNC ) .
(5.3)
Since the eigenvalues Θ((α) λi ) must be congruent to one of the integers 0, 1, . . . , p − 2, p − 1, one can find all the eigenvalues by testing these p numbers. This procedure is carried out until all the eigenvalues and eigenvectors are determined. Let us for the moment assume the correctness of (5.2), and use it in an example by modifying Burnside’s method to use modular integer arithmetic. Solving it is the first step in Dixon’s method, which differs from Burnside’s method at this stage only in using modular arithmetic.
Example 5.1 Solving eigenvalue equations with modular arithmetic Consider the class C2 = {C3 , C3−1 } of the point-group C3v . From (5.2) we know that the class constants λ satisfy the eigenvalue equation H (2) Θ(Λ) = Θ(λ2 ) Θ(Λ) , where the superscripts (α) have been omitted for simplicity. We know the coefficients making up the matrix H (2) from Example 4.6. Now, to solve this in the spirit of Dixon’s method we would first choose a suitable prime √ number. There are six elements in the group C3v so we choose p > 2 6. Let’s pick p = 7. Now we see which integers λ satisfy the eigenvalue equations such that 0 ≤ λ < 7. We could try 0 and find that it does not satisfy the equations. The same would hold for 1, 3, 4, and 5. We know that 2 is an eigenvalue for the class constants and so we should find that 2 satisfies the equations. It does. But it is a doubly degenerate eigenvalue. To avoid the complications arising from degeneracy, when our intent here is simply to illustrate modular arithmetic, let us go on to try λ = 6. A good way to check whether λ = 6 is a satisfactory solution is to use Gaussian elimination with an augmented matrix. The method is as follows. Take the transpose of the secular equation, which gives ) (2) − λI = 0, H where I is the identity matrix. Construct the augmented matrix ⎡ 0−λ 2 0 1 0 ) (2) − λI I = ⎢ H 1 1 − λ 0 0 1 ⎣ 0 0 2−λ 0 0 ) (2) is given by the transpose of H (2) in Example 4.6. where H
⎤ 0 ⎥ 0 ⎦, 1
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Dixon’s method for computing group characters
Now we substitute λ = 6 into the augmented matrix and get ⎡ ⎤ −6 2 0 1 0 0 ⎢ ⎥ 0 0 1 0 ⎦. ⎣ 1 −5 0 0 −4 0 0 1 Well, actually the left section of the augmented matrix is inconsistent with modular arithmetic. We should always keep in mind that when employing modular arithmetic we should confine our integers to the positive realm; we must keep only values 0, 1, . . . , 5, 6. Thus −6 is really +1 mod 7, since 6 + 1 mod 7 = 0 mod 7. We carry out each step just the same way it would be done by computer. The augmented matrix should be ⎡ ⎤ 1 2 0 1 0 0 ⎢ ⎥ ⎣ 1 2 0 0 1 0 ⎦. 0 0 3 0 0 1 Now multiply the first row by 6 mod 7 and add ⎡ 1 2 0 1 ⎢ ⎣ 0 0 0 6 0 0 3 0
it to the second row. This gives ⎤ 0 0 ⎥ 1 0 ⎦ 0 1
for the augmented matrix. The zeroes in the second row of the left-hand side demonstrate that λ = 6 is an eigenvalue. The corresponding eigenvector is given by the second row on the right-hand side, which in column form is ⎡ ⎤ 6 ⎢ ⎥ ⎣1⎦ mod 7. 0 Normalizing the first element of the eigenvector to unity by multiplying each element by 6 (mod 7) gives the eigenvector ⎡ ⎤ 1 ⎢ ⎥ ⎣6⎦ mod 7. 0 One can check that this is a solution as follows: ⎡ ⎤ ⎤⎡ ⎤ ⎡ ⎤ ⎡ 1 6 1 0 1 0 ⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎣2 1 0⎦ ⎣6⎦ = ⎣8⎦ = 6 ⎣6⎦. 0 0 0 0 0 2 Remember! 8 ≡ 1 ≡ 36 mod 7. These procedures are explained in some detail in Appendix 1. The treatment of degenerate eigenvalues is also explained by example.
5.2 Dixon’s method for irreducible characters
99
5.2 Dixon’s method for irreducible characters Consider the character (α) χ(A) of the group element A described by the irreducible matrix ˆ The character is the trace of the matrix, which is invariant under representative (α) Γ(A). unitary similarity transformations. It is convenient and always possible to find a similarity ˆ transformation, U, that diagonalizes the matrix (α) Γ(A): ⎛
U−1
a1 ⎜0 (α) ˆ U=⎜ Γ(A) ⎜ .. ⎝.
0 a2 .. .
0 0 .. .
0 0 .. .
⎞ ··· ···⎟ ⎟ ⎟ ⎠
0
0
···
0
a dα
(5.4)
where dα is the dimension of the Irrep, and the character is given by
(α)
χ(A) =
d α k=1
ak .
(5.5)
Let the order of element A be n(A), so that An(A) = E. Then # $n(A) ˆ = I. ˆn(A) ) = Γ(E) ˆ = Γ(A Γ(A)
(5.6)
The superscripts (α), which designate the Irrep, have been suppressed in order to simplify the notation a bit. We see that the diagonal elements ak of the diagonalized matrix representative satisfy the condition (ak )n(A) = 1. That is, the elements ak are either equal to unity or are roots of unity, and each ak may be expressed in the form ak = ei(2πnk /n(A)) ,
(5.7)
where nk is an integer. However, this does not uniquely define ak since nk = nk + mn(A), with m an integer, is also a satisfactory solution. To uniquely define ak we restrict it to be a primitive root of unity by requiring that
ak = ei(2πnk /n(A)) ,
0 ≤ nk ≤ (n(A) − 1).
(5.8)
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Dixon’s method for computing group characters
The character χ(A) can now be expressed as the polynomial χ(A) =
dα
n(A)−1
ak =
mα (r)ξ r ,
(5.9)
r=0
k=1
where ξ is a primitive n(A)th-root of unity, which means that ξ n(A) = 1, with ξ r = 1, 0 < r < n(A).
(5.10)
The number of times each root ξ r appears along the diagonal is the multiplicity, mα (r). Since the maximum number of times it can appear is given by the dimension of the matrix, we must have 0 ≤ mα (r) ≤ dα . Equation (5.9), which applies to each Irrep (α), can be further simplified. We know that all elements of a class are of the same order and have the same character. Thus we can write that for the class Ci and Irrep (α): χ(Ci ) =
dα
ni −1
ak =
mα (r) ξ r ,
(5.11)
r=0
k=1
where ni is the order of an element belonging to class Ci . A computer program for calculating characters need only use one group element from each class to determine the character of elements of the class. Since each class of a group is of a particular order, we set n be the least common multiple of all the class orders (n1 , n2 , n3 , . . .) of the group. We define n to be the exponent of the group. Thus Rn = E
∀ R ∈ G.
Now let ζ be a primitive nth root of unity. Then ζ n = 1, ζ r = 1,
(5.12) 0 < r < n.
Once again the irreducible characters can be expressed as polynomials in primitive roots of unity with integer coefficients. Now, though, the polynomial is of degree n, which is the exponent of the group. Expressing the character for the class Ci , we obtain
χ(Ci ) =
dα k=1
ak =
n−1 r=0
mα (r) ζ r ,
(5.13)
5.2 Dixon’s method for irreducible characters
101
where the summation over ak is for any element of the class. Note that since n is a multiple of n(R) for all R ∈ G, the equation for χ automatically includes all the necessary primitive n(R)th roots of unity for each element R.
Example 5.2 Characters of C 3v as polynomials in primitive roots C3v has three classes: C1 = {E}, C2 = {D, F }, C3 = {A, B, C},
n(C1 ) = 1, n(C2 ) = 3, n(C3 ) = 2.
D2 = F, D3 = E, A2 = E,
The exponent of the group is n = 6 in this trivial case. Now consider the two-dimensional diagonal matrix Reps for the elements E and A: 1 0 −1 0 E= , A= . 0 1 0 1 The matrices are already in diagonal form, so we can identify the primitive roots of unity by inspection. Using (5.9) we find
n(E)−1
χ(C1 ) = χ(E) =
m(r) ξ r =
r=0
0
m(r) ξ r
r=0
= m(0) ξ 0 = 2 × 1 = 2, since both diagonal elements of E are unity and the multiplicity is obviously 2. For χ(C3 ), we obtain
n(A)−1
χ(C3 ) = χ(A) =
m(r) ξ r =
r=0
1
m(r) ξ r .
r=0
With n(A) = 2, we have ζ = ei(2π/2) = −1. Thus χ(C3 ) = m(0) ξ 0 + m(1) ξ 1 = 1 × (−1)0 + 1 × (−1)1 = 0, because the roots are each of multiplicity 1 by inspection. If we were to use (5.13), which is what we would do in the computer program whose development is outlined in this book, we would use n = 6, the exponent of the group, and obtain, for example, χ(C3 ) = χ(A) =
n−1 r=0
m(r)ζ r =
5 r=0
m(r)ζ r .
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Dixon’s method for computing group characters
The primitive 6th root of unity is ζ = ei(2π/6) = ei(π/3) , giving χ(C3 ) = m(0)ζ 0 + m(1)ζ 1 + m(2)ζ 2 + · · · + m(5)ζ 5 = m(0) + m(1)ei(π/3) + m(2)ei(2π/3) + m(3)eiπ + m(4)ei(4π/3) + m(5)ei(5π/3) . Once again, we can determine terms by inspection, finding that only terms in ζ 0 and ζ contribute, and that m(0) and m(3) are each unity. Thus 3
χ(C3 ) = 1 × e0 + 1 × eiπ = 1 − 1 = 0. The trick of course is to know the multiplicities. If they are known, the calculation of the character is straightforward. Here they were known from inspection. Shortly, we will develop the algorithm for the determination of the multiplicities.
5.2.1 Integer polynomials modulo p We have demonstrated that the characters of group elements can be expressed as polynomials in primitive roots of unity. The roots of unity are in the field of complex numbers, but the coefficients are integers. A key step in the exact calculation of characters is a mapping of these polynomials in the field of complex numbers to the field of integers modulo a prime number p, but with identical integer coefficients modulo p. The mapping is given by
n−1 r=0
Θ
mα (r) ζ r −→
n−1
µα (r) Z r mod p,
(5.14)
r=0
where Z is an integer and µα (r) is an integer modulo a suitably chosen prime number p. Within the constraints given by the prime number p and its modulus, we have
mα (r) = µα (r).
(5.15)
We call the polynomial in Z an integer polynomial modulo p. In the language of pure mathematics, these polynomials form commutative rings. We provide an introduction to these concepts in Appendix 2, along with a proof of the mapping and proofs of the key points that follow in this section. However, for those whose
5.2 Dixon’s method for irreducible characters
103
interest is primarily in using group theory, the outline of the method which follows is quite sufficient. To construct the integer polynomial we seek a prime number p which, for reasons to be demonstrated later, satisfies the conditions √ p>2 g
(5.16)
p = 1 + mn,
(5.17)
and
where g and n are the group order and exponent, respectively, and m is an integer. The existence of such a prime number is demonstrated through Dirichlet’s theorem. Next we define the congruence of two integers u and v, denoted by u ≡ v modulo p
(5.18)
u = v + kp,
(5.19)
as
where k is an integer. For polynomials we define congruence as n i=0
ci xi ≡
n
if ci ≡ di mod p.
di xi mod p,
(5.20)
i=0
Remember, we are considering polynomials with integer coefficients, and it is the corresponding coefficients that are congruent. Now to return to our mapping problem, we replace the primitive root ζ by an integer Z such that
Z n ≡ 1 mod p, Z k ≡ 1 mod p,
(5.21) 0 < k < n.
Z is a primitive root of unity in the field of integers mod p.
Example 5.3 Integer primitive roots √ For our example group, g = 6. Thus, we must choose a prime number p > 2 6. This requires p ≥ 5. But, since the exponent of the group (n = 6) must be a divisor of p − 1,
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Dixon’s method for computing group characters
we must choose p = 1 + 6m = 7, 13, 19, . . . , according to (5.17). Let us take the smallest allowed value, p = 7. Since the exponent of the group is n = 6, we must pick an integer Z such that Z 6 ≡ 1 (mod 7) with Z k ≡ 1 mod 7, k = 1, 2, 3, 4, 5. We find Z by systematic testing, beginning with Z = 2: 21 = 2 22 = 4 23 = 8 ≡ 1 mod 7 Clearly Z = 2 is unsatisfactory. It does not meet the requirement that Z k ≡ 1 mod 7,
0 < k < 7.
Next we test Z = 3: 31 = 3 32 = 9 ≡ 1 mod 7 33 = 27 ≡ 1 mod 7 34 = 81 ≡ 1 mod 7 35 = 243 ≡ 1 mod 7 36 = 729 ≡ 1 mod 7 Thus Z = 3 meets the criteria of (5.16) and (5.17).
Now consider the mapping of (5.14) applied to the character be written as
(α)
Θ
n−1 χi (A) = mαi (r) Z r ,
(α)
χ(A). Clearly, it may
(5.22)
r=0
where i denotes the class to which A belongs and µ has been replaced by m in recognition of the identity of these coefficients as defined by (5.15). The same mapping can be applied to the class constants, namely Θ
(α)
n−1 nCi λi = mαi (r) Z r . dα r=0
(5.23)
5.2 Dixon’s method for irreducible characters
105
Consequently, Θ
(α)
λi
Θ
(α)
nc λj = hijk Θ (α) λk ,
(5.24)
k=1
which, as in the original method of Burnside, leads to the set of eigenvalue equations
H (i) Θ
(α)
Λ = Θ (α) λi Θ (α) Λ .
(5.25)
We have seen how to solve this modular eigenvalue equation in Section 5.1 and more detail is given in Appendix 1.
5.2.2 Multiplicities and the character We have shown that the character can be expressed as a sum of primitive roots of unity. For our present purposes we simplify the notation by dropping subscripts and superscripts that denote a particular class or Irrep and write the character for group element A as
χ(A) =
n−1
mj ξ j .
(5.26)
j=0
Now what we want are the multiplicities, for if we knew them we could determine χ(A) numerically or even express it exactly in polynomial form. We can derive an equation for the coefficients mj , the multiplicities of the roots of unity, following a standard procedure that is quite similar to that for finding the coefficients in a Fourier series expansion. But first, it is convenient to introduce an equation that is similar to (5.26), namely,
χ(At ) =
n−1
mj ξ jt .
(5.27)
j=0
This says that the character of a power of an element is given by a polynomial with the same multiplicities as for the character of the element itself but with each of the roots of unity raised to the same power to which the element is raised. This equation follows from expressing roots of unity as eiθ and recognizing that eiθ1 eiθ2 = ei(θ1 +θ2 ) . Its validity is perhaps best illustrated by example.
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Dixon’s method for computing group characters
Example 5.4 Multiplicities and the character of powers of an element Suppose the unitary diagonal-matrix representation for A is given by ⎞ ⎛ 0 0 eiα ⎟ ⎜ A=⎝ 0 eiα 0 ⎠. iβ 0 0 e Then the character is given by χ(A) = 2eiα + eiβ . Now, the matrix Rep for A2 , for example, is given by ⎛ ⎞⎛ eiα eiα 0 0 0 ⎜ ⎟⎜ 2 iα A =⎝ 0 e eiα 0 ⎠⎝ 0 0 0 eiβ 0 0 ⎛ ⎞ e2iα 0 0 ⎜ ⎟ =⎝ 0 e2iα 0 ⎠, 0 0 e2iβ
⎞ 0 ⎟ 0 ⎠ iβ e
and the character is given by χ(A2 ) = 2e2iα + e2iβ , with the same multiplicities as for the character of A itself, as expected from (5.27). From the modular class constants, we can compute the modular characters Θ according to Θ
(α)
dα χi = Θ (α) λi , nCi
(α)
χi
(5.28)
if we know the dimension of the Irrep. Using the modular equivalent of (4.52), we obtain the dimensions of the different Irreps from |dα |2 ≡ i
g mod p. Θ (λi ) Θ λ−1 /nc(i) i
(5.29)
2 √ Since g = dα , the choice of p > 2 g ensures that p > 2dα so that we can determine the dα s uniquely.
5.3 Computer codes for Dixon’s method Now, knowing Θ relation
(α)
χi
and Z, we can compute the multiplicities mαi (r) from the
mαi (r) =
=
n−1 1 Θ (α) χi (At ) Z −rt mod p n t=0 n−1 1 n t=0
(α)
χi (At ) ζ −rt .
The proof of this equation is n−1 1 n t=0
107
(α)
t
χi (A ) ζ
−rt
n−1 1 = n t=0
n−1
(5.30)
mαi (u) ζ
ut
ζ −rt
u=0
=
n−1 n−1 t(u−r) 1 ζ mαi (u) n u=0 t=0
=
n−1 1 mαi (u) n δu,r n u=0
= mαi (r),
(5.31)
where we have used the fact that the group element At has an eigenvalue ζ ut occuring with the same multiplicity mαi (u) as the eigenvalue ζ t of the group element A, as shown by (5.29). Since 0 ≤ mαi (r) ≤ dα ≤ p , (5.30) determines the multiplicities mαi (r) uniquely. Finally, one uses (5.28) to determine the irreducible characters, as demonstrated in Appendix 2. 5.3 Computer codes for Dixon’s method Dixon’s method is readily amenable to computational techniques. Numerical algorithms for modulo arithmetic as well as the Gauss elimination method are easily implementable. Moreover, there are existing packages that can simplify further the implementation of Dixon’s method, foremost among those is found in Mathematica. Mathematica provides a computational package for number theory that contains many useful functions for this purpose. It is called NumberTheory‘NumberTheoryFunctions‘. The functions that are relevant here are: • • • • • •
LCM[integer list] : gives the least common multiple of the integer list. Prime[n] : gives the nth prime number. Mod[m,n] : gives the remainder on division of m by n, i.e. m modulo n. PowerMod[a,b,n] : gives ab modulo n. PowerMod[a,-1,n] : gives the inverse of a modulo n. MultiplicativeOrder[k,n] : gives the smallest integer m such that km ≡ 1 modulo n.
Dixon’s method for computing group characters
108
Generating group exponent ngrp First, we generate the exponents of representative class elements, then find the lowest common multiple among them. The exponent of class i is found by satisfying the relation LC[i, 1]Cexp[i] = E, the identity element; where LC[i,1] is the first element in the ith class array, and Cexp[i] is the array containing class exponents. m[i,j], the binary product of elements i and j, is used in the multiplication process.
Cexp={1} Do[ t=LC[i,1];ncexp=1; While[ t =1,t=m[[LC[[i,1]],t]];ncexp==ncexp+1 ]; AppendTo[Cexp,ncexp],{i,2,nc} ] ngrp=Cexp[[2]] Do[ ngrp=LCM[ngrp,Cexp[[i]]],{i,3,nc} ]
Generating the prime number pp i=2; pp=Prime[2] While[Mod[pp,ngrp]!=1],i++;pp=Prime[i]]
Generating the primitive root Z Z=2; mm=1 While[mm!=ngrp,Z++;mm=MultiplicativeOrder[Z,pp]]
Gauss’s method for eigenvalue problem We denote the class-matrices by CM[ii,i,j], where ii is the class index and i,j the matrix element indices. We first set up the augmented matrix, which is composed of the transpose of CM[ii], together with the augmenting matrix augm[i,j], such that eigenvectors will appear as eigenrows of augm[i,j]. Using -ev as trial eigenvalue, which can assume integer numbers between 0 and (pp-1), we determine the class-constant eigenvectors of CM[ii], which now appear as rows of augm[i,j]. If degenerate eigenspaces are present, we repeat this procedure, but now we use augm[i,j] composed of the eigenrows of the class-matrix CM and augment
5.3 Computer codes for Dixon’s method
109
it with another class-matrix, say CM , after projecting out the degenerate eigenspace of augm[i,j]. We carry out the search for eigenrows only on the degenerate subspace. This process is repeated until all nondegenerate eigenvectors of the class-constant matrices are obtained. Current eigenrows are stored in Dirac: 1. Initialize augm[i,j] as eigenrows of the identity class-matrix CM[[1]], with degeneracy nc. nc2=2*nc ii=1
Do Dirac[i,j]=0,{i,1,nc},{j,1,nc}
Do last[i]=0,{i,1,nc} last[1]=nc
Do
Do augm[i,j]=0,{j,1,nc} ; augm[i,i]=1,{i,1,nc} 2. Construct the matrix c[i,j]= k augm[i,k]*CMTranspose [k,j], where CM is the current class-matrix, and i runs over the rows of the degenerate subspace, which start with row ist and end at row ind=last[ist]. While[ ii
Do
Do c[i,j]=0;
Do c[i,j]+=augm[i,k]*b[[k,j]], {k,1,nc} , {j,1,nc} , {i,ist,ind} ; 3. Start trial eigenvalues ev, from 0 to (pp-1). Project out the eigenspace of known eigenvectors by constructing the matrix, evd= augm*CM(transpose)-λ*augm, which is just
Dixon’s method for computing group characters
110
the subspace to the eigenvectors in augm. Construct the augmented matrix as orthogonal
d= augm evd , and convert it to modulo pp. Use flag to give the degeneracy of the eigenvalue.
ev=-1; While[ And[ev
Do
evd[i,j]=ev augm[i,j] + c[i,j], {i,1,nc},{j,1,nc} ;
Do
Do
Do
k=j+nc;d[i,j]=augm[i,j]; d[i,k]=evd[i,j],{j,1,nc} , {i,1,nc} ;
d[i,j]=Mod[d[i,j],pp],{i,1,nc}, {j,nc+1,nc2} ; ix=ven;
4. Begin Gauss elimination, in modulo pp, over the degenerate subspace that starts at row ist, and ends at row ind. When a row of zeroes is found in evd, then the corresponding row in augm is an eigenrow. The new eigenrow is then stored in Dirac.
Do
jj=nc+1; While[And[d[i,jj]==0,jj<=nc2],jj++]; Switch[jj<=nc2, True, inv=PowerMod[d[i,jj],-1,pp];
Do d[i,j]=Mod[inv*d[i,j],pp], {j,1,nc2} ;
5.3 Computer codes for Dixon’s method
Do
111
dd=pp-d[k,jj];
Do d[k,l]=Mod[dd*d[i,l] +d[k,l,pp], {l,1,nc2} , {k,i+1,ind} ,
False,flag++;ven++;jx=ven-1;
Do Dirac[jx,j]=d[i,j], {j,1,nc} ; ],{i,ist,ind} ]; last[ix]=jx ];
5. At the end of the eigenvalue search Dirac is used to update the augm matrix with the newly determined eigenrows.
Label[Fin];
Do augm[i,j]=Dirac[i,j], {i,ist,ind}, {j,1,nc} ;
6. If all nondegenerate eigenvectors have been obtained, exit the eigenvector search. Otherwise, continue reducing the remaining degenerate subspaces.
Switch[ last[ist]==nc, True,i=0; While[i
Dixon’s method for computing group characters
112
Switch[ qq1==1,True,Goto[Gd] ], False,ist=last[ist]+1; Switch[ last[ist]==0, True, Goto[Panic], False,Goto[Try] ] ];Label[Begin] ]
7. The final nondegenerate eigenrows are stored in Dirac. Normalize the eigenrow matrix Dirac so that the first column is composed of ones. Each row of Dirac will then contain an Irrep Dirac characters in modulo pp. Label[Gd]
Do Inv1=PowerMod[Dirac[i,1],-1,pp]
Do Dirac[i,j]=Mod[Dirac[i,j]*inv1,pp], {j,1,nc} ],{i,1,nc} ] 8. Use the Dirac characters to compute the dimensions of the Irreps d(i), in modulo pp, using the relation 2 d2 (i) = g/ λ (j)/r(j) , where λ(j) is the Dirac character of the jth class, and r(j) the class size.
Do dim2[i]=0,{i,1,nc}
Do d2=0;
Do ri=PowerMod[Length[LC[[j]]],-1,pp];
5.3 Computer codes for Dixon’s method
113
d2=Mod[d2+Dirac[i,j]**2*ri,pp], {j,1,nc} ]; d2=PowerMod[d2,-1,pp]; dm=Mod[g*d2,pp]; dim2[i]=dm,{i,1,nc}
]
Do
Switch[ dim2[i]==1, False,j=1;dm=1; While[And[j
n−1
Θ
(α)
χi (At )
Z −rt (mod p)
t=0
where α and i are Irrep and class indices, respectively, and Z is the primitive root modulo pp.
cip={} chit={}
Do
Do ri=PowerMod[Length[LC[[j]]],-1,pp]; ch=Mod[ri*dim2[[i]]*Dirac[[i,j]],pp]; AppendTo[chit,ch],{j,1,nc} ]; AppendTo[cip,chit];chit={},{i,1,nc} ]
Dixon’s method for computing group characters
114
nm1=PowerMod[nn,-1,pp]
Do mult[i,j,k]=0,{i,1,nc},{j,1,nc},{k,1,nn}
Do mmx=0;
Do mmx+=cip[[i,cc1[j,l]]]*zz[k,l]; mult[i,j,k]=Mod[mmx*nm1,pp],{l,1,nc} ], {i,1,nc},{j,1,nc},{k,1,nn} ] 10. Use the multiplicities to compute the Irrep characters, according to the relation (α)
χ(i) =
n−1
mαi (j)ξ j
j=0
and store them in char.
Do char[i,j]=0,{i,1,nc},{j,1,nc}
Do
Do char[i,j]+=mult[i,j,k]*Exp[i(2*π*(k-1)/nn), {k,1,nn} ], {i,1,nc},{j,1,nc} ]
11. Set up Irreps in ascending order of their dimensions. Store the complete onedimensional Irreps of group elements in file Γ, and the corresponding class characters in array charac. The first Irrep is the identity.
tempo={} Γ = {}
Do charac[i,j]=0,{i,1,nc},{j,1,nc}
Do ban[i]=0;dim[i]=0,{i,1,nc}
5.3 Computer codes for Dixon’s method
115
dim[1]=1 nrep=1
Do AppendTo[tempo,1],{j,1,nc} ;
Do charac[1,j]=1,{j,1,nc} ; AppedTo[Γ,tempo];tempo={}
Do Switch[ And[dim2[i]==1,ban[i]==0, Sum[char[i,j],{j,1,nc} !=nc], True,nrep++;ban[i]=1; dim[nrep]=dim2[i];
Do AppendTo[tempo,charac[i,indc[j]]], {j,1,g} ; AppendTo[Γ,tempo];tempo={};
Do charac[nrep,j]=char[i,j],{j,1,nc} ],{i,1,nc} ]
12. Add the characters of the remaining Irreps to the array charac, in ascending order of their dimensions.
i1=1; i3=nrep+1
√ dmax=ItegerPart[ g While[
And[i1
Dixon’s method for computing group characters
116
Do charac[i3,j]=char[i,j], {j,1,nc} ; i3++ ]; i++ ] ]
13. Print the Irrep characters in ascending order of dimensions.
chi=Array[charac,{nc,nc}
Print["IRREPS PRINTED IN ASCENDING ORDER OF DIMENSION: "] Print[IRREP DIMENSION: dim= ",
dimension=Array[dim,{nc} ]
Print["IRREDUCIBLE CHARACTERS: "]
Do Print[StyleForm["χ(",FontWeight→ "Bold"], i,"] = " Simplify[chi[[i]]]],{i,1,nc}
Label[Panic]
Appendix 1 Finding eigenvalues and eigenvectors The basis for the computational methods that are central to computational group theory as described in this book is the solution of the eigenvalue equation (α)
H (i) Λ(α) = λi Λ(α) ,
(5.32)
where α denotes an Irrep and i denotes the ith class. We know from Chapter 5 how to find the set of coefficients hijk that make up the elements of the matrix H (i) . The unknowns (α) are the eigenvalues λi and the eigenvectors Λ(α) . Suppose the matrix H (i) is 3 × 3. Then we must find three eigenvalues and three corresponding eigenvectors as represented by three equations, one for each irreducible
Appendix 1 Finding eigenvalues and eigenvectors
117
representation: (1)
H (i) Λ(1) = λi Λ(1) (2)
H (i) Λ(2) = λi Λ(2) (3)
H (i) Λ(3) = λi Λ(3) We can represent this set of equations in matrix form as H (i) V = V λ,
(5.33)
where the columns of the matrix V are the eigenvectors Λ ⎡ (1) ⎤ λi 0 0 ⎢ ⎥ (2) λ=⎣ 0 λi 0 ⎦. (3) 0 0 λi
(α)
and λ is the matrix
We can rewrite (5.33) and set its determinant equal to zero, giving (i) H V − V λ = 0. Of course the eigenvectors are unknown, so we will each dimension, as represented by the unit matrix. ⎡ 1 0 0 ⎢ V =⎣ 0 1 0 0 0 1
(5.34)
start with three basis vectors, one for Thus we take ⎤ ⎥ ⎦.
as a starting set of basis vectors that span the three-dimensional space. It is important to think of V as a matrix of vectors spanning a space, not simply as the unit matrix. The importance of this distinction will be made clear in a later section on Dixon’s method, where it will have important implications for solving (5.33) in the case of degenerate eigenvalues. Now we must address the problem of the eigenvalues. Suppose we could guess them. There are ways of systematically iterating an original guess so as to converge on the true value, but here we mean an exact guess. Of course, in general, this would be a pointless procedure. One would usually have zero probability of guessing correctly from a continuous set of possible eigenvalues. On the other hand, if the eigenvalues were known to be a subset of a reasonably sized finite set of known possible values, which is the case for Dixon’s method, guessing might have an advantage over other methods. More accurately, rather than making guesses, one could simply try all possible eigenvalues and find which ones satisfy (5.34).
Example 5.5 Consider the matrix
H=
4 2
1 3
,
118
Dixon’s method for computing group characters
and suppose the possible eigenvalues are known to be integers 0 ≤ λ ≤ 10. We could then just substitute trial values of λ into the secular equation 1 4−λ =0 2 3−λ to find the actual values. One finds from direct substitution of trial values that λ = 2, 5. However, there is a more systematic way of checking, a way that in the process also yields the corresponding eigenvectors. Write down an augmented matrix for which the left-hand side (l.h.s.) consists of the elements in the determinant of (5.34) and the right-hand side (r.h.s.) is the matrix V . For our present example we have 4−λ 1 1 0 . 2 3−λ 0 1 Now the l.h.s. of the augmented matrix will be zero if any column (or row) consists of all zeroes. Thus we can substitute possible values of λ and systematically operate on columns by standard procedures to see if it is possible to get a column of zeroes in the l.h.s. At the same time, we make identical operations on the corresponding columns in the r.h.s. of the augmented matrix. To see how this works, we try one of the known eigenvalues, λ = 2. The augmented matrix becomes 2 1 1 0 2 1 0 1 for which the operations are trivial. Multiply the first column of each side of the augmented matrix by 1/2. Then, on each side of the augmented matrix, subtract the first column from the second column. The result is 1 0 1/2 −1/2 . 1 0 0 1 The column of zeroes, the second column on the l.h.s., tells us that the equations are satisfied and that λ = 2 is indeed an eigenvalue. The corresponding column on the r.h.s. is now the eigenvector belonging to the eigenvalue λ = 2. One can check that this is so by substituting into (5.32), obtaining 4 1 −1/2 −1/2 =2 , 2 3 1 1 carrying out the operation, and verifying that the solution is correct. Now, in actual practice, especially when using a computer program, it is often more convenient to take the transpose of the matrices and operate on the rows of the corresponding augmented matrix to try to get a row of zeroes on the l.h.s. If that is possible, the remainder of the row on the r.h.s. contains the corresponding eigenvector.
Appendix 1 Finding eigenvalues and eigenvectors
119
The Gauss elimination method: a peculiar method for matrix eigenvalue problems! The essence of Burnside’s method is that the class constants can be determined by diagonalizing the class-constant matrix and then finding the group characters from the class constants, as described in Chapter 4. However, rather than try to directly diagonalize a class-constant matrix, we now work through an example problem to find the class constants for the group C3v using the procedure we have just described for finding eigenvalues and eigenvectors. We are doing it here only for tutorial purposes, as a preliminary to Dixon’s method, but without yet introducing the complications of modular arithmetic that are an ingredient of Dixon’s method. This procedure finds its roots in the method of Gauss elimination, which is traditionally applied to the matrix inversion problem. The inverse of a matrix A is found through the defining equations A X = l,
(5.35) −1
lX=A where l is the identity matrix. This can ⎡ a11 a12 ⎢ ⎣ a21 a22 a31 a32 ⎡
1
⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎣ 0
,
(5.36)
be written in augmented matrix form as ⎤ a13 1 0 0 ⎥ a23 0 1 0 ⎦, a33 0 0 1
0
a−1 11
a−1 12
a−1 13
1
0
a−1 21
a−1 22
a−1 23
0
1
a−1 31
a−1 32
a−1 33
0
(5.37)
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦
(5.38)
for the case of a 3 × 3 matrix. The steps from equation (5.37) to equation (5.38) follow the Gauss elimination recipe. Now, consider the matrix eigenvalue problem $ # XT AT − λ l = 0, (5.39) where we have taken the transpose of A, for reasons to be clarified shortly. If λ is replaced by ω = λ, then, following the Gauss elimination method we have $ # XT AT − ω l = l, (5.40) # $−1 lX = AT − ω l . (5.41)
However, if now we put λ back, satisfying (5.39), the singularity of AT − ω l implies that one or more of its rows are linear combinations of one or more of the remaining rows. The process of the Gauss elimination will then produce zero rows, the number of which will reflect the degeneracy of the eigenvalue λ. For the sake of simplicity, we use the transpose of A so that the eigenvectors appear as eigenrows. Upon carrying out Gauss row elimination,
120
Dixon’s method for computing group characters
we find that for every row of zeroes that appear on the l.h.s. of the augmented matrix, the corresponding row on the r.h.s. is just the eigenvector that produces the null space represented by the zero row. This scenario assumes that we know the actual eigenvalues of A. As we demonstrated, when we work in modulo a prime number p, the eigenvalue has to be one of the integers 0 < i < p. In this case, as we try the different integers, only true eigenvalues will produce zero rows. Usually, the Gauss elimination method is used to find the solution of a system of linear equations or to find the inverse of a matrix. Since it is usually the eigenvalues that are sought, this method is never used in a conventional matrix eigenvalue problem.
Implementation of Gauss elimination in Burnside’s method In constructing the class-constant matrices with a computer program, the form of the matrices will vary depending upon the choice of generators and the order in which they are read into the program. One possibility for the matrix H (3) is ⎤ ⎡ 0 1 0 ⎥ ⎢ H (3) = ⎣ 3 0 3 ⎦. 0 2 0 Compare this with H (3) from Chapter 4. Now we systematically operate on rows of the augmented matrix but using the transpose of H (3) and other matrices to determine eigenvalues and eigenvectors. The augmented matrix is ⎤ ⎡ −λ 3 0 1 0 0 # $ ⎥ ˜ (3) −λV˜ V˜ = ⎢ 2 0 1 0 ⎦. V˜ H ⎣ 1 −λ 0 3 −λ 0 0 1 Note the change in order for matrix multiplication because of the use of transposed matrices. We have used λ as a scalar because we are going to check possible values of λ one at a time. For expediency we substitute λ = 3, a known eigenvalue, to demonstrate a systematic procedure. This gives ⎡ ⎤ −3 3 0 1 0 0 ⎢ ⎥ 2 0 1 0 ⎦. ⎣ 1 −3 0 3 −3 0 0 1 Operate on the first row and multiply the first nonzero element by −1/3, its negative inverse. Multiply all other elements of the first row by −1/3. This makes +1 the first nonzero element of row one and gives ⎡ ⎤ 1 −1 0 −1/3 0 0 ⎢ ⎥ 2 0 1 0 ⎦ ⎣ 1 −3 0 3 −3 0 0 1 for the augmented matrix.
Appendix 1 Finding eigenvalues and eigenvectors
121
Row operations, like the one we have just described and performed, can be accomplished by matrix multiplication. For example, to multiply the first row by −1/3 is identical to operating with the matrix ⎤ ⎡ −1/3 0 0 ⎥ ⎢ M1 = ⎣ 0 1 0 ⎦. 0 0 1 Now go to the second row. If the element in the second row just below the first +1 in the first row is nonzero, make it zero. Here, multiply the first row by −1 and add it to the second row. This gives ⎡ ⎤ 1 −1 0 −1/3 0 0 ⎢ ⎥ 2 1/3 1 0 ⎦. ⎣ 0 −2 0 3 −3 0 0 1 Here we have implicitly operated with the matrix ⎤ ⎡ 1 0 0 ⎥ ⎢ M2 = ⎣ −1 1 0 ⎦. 0 0 1 Now we note that the sequence of matrix operations we have performed thus far, namely, ⎤ ⎤ ⎡ ⎤⎡ ⎡ −1/3 0 0 −1/3 0 0 1 0 0 ⎥ ⎥ ⎢ ⎥⎢ ⎢ M = M2 M1 = ⎣ −1 1 0 ⎦ ⎣ 0 1 0 ⎦ = ⎣ 1/3 1 0 ⎦. 0 0 1 0 0 1 0 0 1 is identical to the r.h.s. of the augmented matrix. Consider the implications of what we are doing. We are making a systematic sequence of matrix multiplications with the aim of achieving a row of zeroes, if we have correctly chosen an eigenvalue. Simultaneously, we are operating on a set of basis vectors represented by the matrix V . These matrix operations stretch, reflect, or rotate the basis vectors. If we have correctly chosen an eigenvalue, and have demonstrated that the eigenvalue equation is satisfied, we have also transformed the set of basis vectors such that at least one of them is now an eigenvector. Let us now continue with this systematic procedure. Go to the third row, same column, to make that element zero by adding to it the appropriate multiple of the first row. Here the element is already zero, so we start over with this general procedure beginning with the second row and second column. If that element is zero, move to the third column. That is, go to the first nonzero element in the second row. Here that is −2 in the second column. Multiply the second row by the inverse of −2 to get ⎡ ⎤ 1 −1 0 −1/3 0 0 ⎢ ⎥ 1 −1 −1/6 −1/2 0 ⎦. ⎣ 0 0 3 −3 0 0 1
122
Dixon’s method for computing group characters
Now multiply the second row by −3 and add it to the third row, so as to make the second column of the third row equal to zero. This gives ⎡ ⎤ 1 −1 0 −1/3 0 0 ⎢ ⎥ 1 −1 −1/6 −1/2 0 ⎦. ⎣ 0 0 0 0 3/6 3/2 1 The zeroes in row 3 on the l.h.s. indicate that the trial value of λ = 3 is an eigenvalue and that the r.h.s. of row 3 is the corresponding eigenvector. As discussed in Chapter 4, we now normalize the eigenvector by making the first element equal to unity. We do this in the present case by multiplying all components of the eigenvector by 2. The solution is then ⎤ ⎡ 1 ⎥ ⎢ λ = 3, Λ = ⎣ 3 ⎦. 2 This can be checked by applying ⎡ 0 ⎢ ⎣ 3 0
H(3) to Λ = [1, 3, 2], namely, ⎤ ⎤ ⎡ ⎤⎡ 1 1 1 0 ⎥ ⎥ ⎢ ⎥⎢ 0 3 ⎦ ⎣ 3 ⎦ = 3 ⎣ 3 ⎦. 2 2 2 0
In a similar manner, one can demonstrate that the other two solutions are ⎤ ⎤ ⎡ ⎡ 1 1 ⎥ ⎥ ⎢ ⎢ Λ = ⎣ 0 ⎦. λ = −3, Λ = ⎣ −3 ⎦ and λ = 0, −1 2
Modular arithmetic The key point of Dixon’s method is that the class constants, which are the eigenvalues of (1), are integers modulo an appropriately chosen prime number. It is that feature which makes it practical to systematically apply the Gauss elimination procedure to test trial solutions to (5.1). As a preliminary to demonstrating Dixon’s method, we present here a few definitions and some techniques for working with integers in modular arithmetic. Two integers k and l are said to be congruent modulo p if they differ by an integer multiple of the integer p. For Dixon’s method, p is further restricted to being a prime number. The congruence is written k ≡ l mod p and p is called the modulus of the congruence. Congruent numbers are, roughly speaking, equal. The most familiar use of congruence is in specifying the time of day. We specify hours up to 12 and then start over again. The hands on the clock specify the time modulo 12. When adding or multiplying numbers (mod p) we must restrict the resulting number to the set of integers 0, 1, . . . , p − 1. We do this, if necessary, by replacing the number
Appendix 1 Finding eigenvalues and eigenvectors
123
generated by its equivalent from the set of integers 0, 1, . . . , p − 1. This is convenient to do with a Fortran program by making use of the function mod(m, p).
Example 5.6 Consider the integer 19 mod 7. mod(19, 7) = 19 − INT
19 7
× 7 = 19 − (2 × 7) = 5.
On the other hand, the integer i = −19 mod 7, which we might expect to become −5, actually becomes 2. Integers are always shifted by an integer multiple of p in order to bring them into the range 0 ≤ i < p.
In modular arithmetic, the sum, difference, product, or quotient of integers gives another integer such that when it is divided by p the remainder is an integer r, 0 ≤ r < p. This has an important effect on the way we determine eigenvalues in Dixon’s method. Matrix eigenvalue problem in Dixon’s method We now have the tools at hand to determine the eigenvalues of (5.32) using Dixon’s method and modular arithmetic. We begin as before by constructing an augmented matrix, only this time we use +λ on the l.h.s., not −λ , and we use a prime on λ. This means that the actual eigenvalues λ will be the negative (mod p) of the values λ we determine. We do this because with modular arithmetic all answers will be integers 0 ≤ λ < p, where p is a prime number such that √ p > 2 g. In the present case we set p = 7. It is more convenient to have all positive integers in the augmented matrix at all stages. With +λ in the augmented matrix we have ⎤ ⎡ λ 3 0 1 0 0 ⎥ ⎢ 2 0 1 0 ⎦ mod 7 ⎣ 1 λ 0 3 λ 0 0 1 For comparison with what we did with Burnside’s method, we substitute λ = 3 to demonstrate Dixon’s method. This gives ⎤ ⎡ 3 3 0 1 0 0 ⎥ ⎢ ⎣ 1 3 2 0 1 0 ⎦ mod 7 0 3 3 0 0 1 Operate on the first row and multiply the first nonzero element by its inverse. The first nonzero element is M11 = 3 and its inverse (mod 7) is 5. Multiply all other elements of the first row by 5. This gives ⎤ ⎡ 1 1 0 5 0 0 ⎥ ⎢ ⎣ 1 3 2 0 1 0 ⎦ mod 7 0 3 3 0 0 1 for the augmented matrix.
124
Dixon’s method for computing group characters
Now go to the second row. If the element in the second row just below the first 1 in the first row is nonzero, make it zero. Here, that is the element M21 = 1. We must add to it the integer 6 to get M21 ≡ 7 ≡ 0. We can accomplish this by multiplying each element of the first row by 6 and adding it to the corresponding element of the second row to get ⎡ ⎤ 1 1 0 5 0 0 ⎢ ⎥ ⎣ 0 2 2 2 1 0 ⎦ mod 7 0 3 3 0 0 1 Next go to the third row, same column, to make that element zero by adding to it the appropriate multiple of the first row. Here the element M31 is already zero, so we start over with this general procedure beginning with the second row and second column. If that element is zero, move to the third column. That is, go to the first nonzero element in the second row. Here that is M22 = 2. Multiply the second row by the inverse of 2, which is 4, to get ⎤ ⎡ 1 1 0 5 0 0 ⎥ ⎢ ⎣ 0 1 1 1 4 0 ⎦ mod 7 0 3 3 0 0 1 Now multiply the second row M32 = 0. This gives ⎡ 1 ⎢ ⎣ 0 0
by 4 and add it to the third row, so as to make the
1 1 0
0 1 0
5 1 4
0 4 2
⎤ 0 ⎥ 0 ⎦ mod 7 1
The set of zeroes in row 3 on the l.h.s. indicates that the trial value of λ = 3 is a solution and that the r.h.s. of the row is the corresponding eigenvector, but the eigenvalue is actually the negative of λ , that is, λ = −λ . We can verify by direct substitution that the eigenvalue λ ≡ −3 ≡ 4 mod 7 and its corresponding eigenvector satisfy the eigenvalue equation, (5.32) We find ⎡
0 ⎢ ⎣ 3 0
1 0 2
⎤ ⎤ ⎡ ⎤ ⎡ ⎤⎡ 4 2 4 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥⎢ 3 ⎦ ⎣ 2 ⎦ ≡ ⎣ 1 ⎦ ≡ 4 ⎣ 2 ⎦ mod 7 1 4 1 0
Similarly, we can find that the remaining two solutions are ⎤ ⎤ ⎡ ⎡ 4 6 ⎥ ⎥ ⎢ ⎢ and λ = 3, Λ = ⎣ 5 ⎦ mod 7 λ = 0, Λ=⎣ 0 ⎦ 1 1 We have found that λ ≡ 4 mod 7 is an eigenvalue with eigenvector [4 1 0] (in row form). The eigenvector needs to be normalized such that the first component, which corresponds to the identity class, must equal unity mod 7. This can be accomplished by multiplying each component of the eigenvector by 2 mod 7, which is the inverse of 4, the first
Appendix 1 Finding eigenvalues and eigenvectors
125
component. A similar procedure for the remaining two eigenvectors gives the following normalized solutions. ⎤ 1 ⎥ ⎢ Λ = ⎣ 4 ⎦; 2
λ = 4,
⎤ 1 ⎥ ⎢ Λ = ⎣ 0 ⎦; 6
⎤ 1 ⎥ ⎢ Λ = ⎣ 3 ⎦ mod 7 2 ⎡
⎡
⎡
λ = 0,
λ = 3,
From inspection of these results, the reader will note that the three eigenvalues correspond to the class constants in the second row of each of the eigenvectors.
Degenerate eigenvalues It may happen that the class-constant matrix yields degenerate eigenvalues. For example, if we had used a different set of generators, or if we used the same generators but introduced them in a different order into a computer program, the first nontrivial class-constant matrix, that is, the first class-constant matrix generated other than that for the identity class, could have been ⎤ ⎡ 0 1 0 ⎥ ⎢ H (2) = ⎣ 2 1 0 ⎦ mod 7 0 0 2 The transposed augmented matrix is then ⎡ λ 2 # $ ⎢ (2) ˜ ˜ ˜ ˜ = ⎣ 1 1 + λ VH +λ V V 0 0
0 0 2 + λ
1 0 0
0 1 0
⎤ 0 ⎥ 0 ⎦ mod 7 1
We would now try values of λ beginning with λ = 0 to see if our systematic sequence of row operations would identify λ as a solution. We would find that the first solution is obtained for λ = 1, for which the augmented matrix after making the appropriate row operations becomes ⎡ ⎤ 1 2 0 1 0 0 ⎢ ⎥ ⎣ 0 0 0 6 1 0 ⎦ mod 7 0 0 3 0 0 1 Since the actual eigenvalues λ are the negative of λ mod 7, the solution is ⎤ ⎡ 6 ⎥ ⎢ λ = 1, λ = 6, Λ = ⎣ 1 ⎦ mod 7 0
(5.42)
Proceeding on, we find that there are no further solutions until λ = 5, for which the multiplicity is 2 and the augmented matrix after having made the appropriate row operations is ⎡ ⎤ 1 6 0 3 0 0 ⎢ ⎥ ⎣ 0 0 0 4 1 0 ⎦ mod 7 0 0 0 0 0 1
126
Dixon’s method for computing group characters
The two eigenvectors belong to the same eigenvalue. They span the null space belonging to λ = 5, or λ = 2 (mod 7) and they also define a two-dimensional vector space, a plane. Because the eigenvectors belong to a doubly degenerate eigenvalue, any linear combination of the two is also an eigenvector. One can check this by taking, say, the direct sum of [4 1 0] and [0 0 1], which is [4 1 1], and showing that H (2) operating on this direct sum gives back the same eigenvector multiplied by the eigenvalue λ = 2. However, we seek a unique set of eigenvectors. Remember, each eigenvector corresponds to a particular Irrep. There are three of them in our example problem and each is unique. Thus we know there are two other eigenvectors in the same two-dimensional space that are linear combinations of the two we have just found, but which are unique in the following sense: Both belong to the doubly degenerate eigenvalue, λ = 2, for the eigenvalue equation H (2) Λ = λ2 Λ, but each of the two eigenvectors belongs to a different (nondegenerate) eigenvalue when the eigenvalue equation to be solved is for a different class-coefficient matrix. These particular eigenvectors are the ones we seek. What we are doing (or trying to do) is equivalent to diagonalizing a matrix. When eigenvalues are degenerate it is not always possible to diagonalize the matrix. It is the curious combination of eigenvectors and eigenvalues which are inextricably connected that makes it all possible here. To reiterate, we solved a particular eigenvalue problem and found one of the eigenvectors. By projecting out this eigenvector we reduced the vector space to two dimensions. We then found a doubly degenerate solution to the eigenvalue equation. We now move on to a new eigenvalue problem for which two new nondegenerate eigenvectors are found that are some linear combination of the two original eigenvectors that were degenerate when working with H (2) . The procedure at this stage is to begin anew with the next class-constant matrix, but retain the information we have learned about the eigenvectors. The algorithm that yielded the class-constant matrix H (2) that was just used would now yield ⎤ ⎡ 0 0 1 ⎥ ⎢ H (3) = ⎣0 0 2⎦ 3 3 0 We again construct the augmented matrix # ˜ (3) + λ V˜ V˜ H
$ V˜ mod 7
but this time we take for our starting value of V˜ a matrix whose rows include the set of degenerate basis vectors we have just found, namely, ⎤ ⎡ 6 1 0 ⎥ ⎢ V˜ = ⎣4 1 0⎦ mod 7 0 0 1 The rows are the set of eigenvectors we have just found using the class-constant matrix ˜ , as H (2) . It does not matter in which order the eigenvectors are placed in the rows of W the reader can check by duplicating the following steps with a different ordering of the rows. We place the unique eigenvector in row 1 for convenience. We will soon eliminate it.
Appendix 1 Finding eigenvalues and eigenvectors
127
The general transposed augmented matrix for H (3) becomes ⎡
6λ ⎢ ⎣ 4λ 1
λ λ 2
21 15 λ
6 4 0
1 1 0
⎤ ⎡ 6λ 0 ⎥ ⎢ 0 ⎦ ≡ ⎣ 4λ 1 1
λ λ 2
0 1 λ
6 4 0
1 1 0
⎤ 0 ⎥ 0 ⎦ mod 7 1
We again check trial values for the eigenvalues, beginning with λ = 0. For λ = 0 the augmented matrix becomes ⎡
0 ⎢ ⎣ 0 1
0 0 2
6 4 0
21 15 0
1 1 0
⎤ ⎡ 0 0 ⎥ ⎢ 0 ⎦≡⎣ 0 1 1
0 0 2
0 1 0
6 4 0
1 1 0
⎤ 0 ⎥ 0 ⎦ mod 7 1
The l.h.s. of the first row is already zero, and the r.h.s. of the row contains a known eigenvector belonging to a nondegenerate eigenvalue, that is, an eigenvalue of multiplicity one. Thus we must restrict our row operations to rows 2 and 3, which define a twodimensional vector space containing the new eigenvalues and associated eigenvectors that we seek. The augmented matrix is then
4λ 1
λ 2
1 λ
4 0
1 0
0 1
mod 7
(5.43)
In this case, we would find that λ = 0 is not an eigenvalue, so we move on to other trial values. Carrying out row operations as before, with modular arithmetic, we find that solutions exist for λ = 3 and λ = 4. The actual eigenvalues are the negative of these mod 7, so the solutions are found to be λ = 3 λ =4
*
⎡ ⎤ 2 ⎢ ⎥ Λ = ⎣4⎦ ; 1
λ = 4 λ =3
*
⎡ ⎤ 5 ⎢ ⎥ ⎣3⎦ mod 7 1
(5.44)
We can normalize the eigenvectors from (5.44) and combine them with the normalized version of (5.42) to get the following solution for the modular class constants of C3v :
λ = 6,
⎡ ⎤ 1 ⎢ ⎥ Λ = ⎣6⎦ ; λ = 4, 0
⎡ ⎤ 1 ⎢ ⎥ Λ = ⎣2⎦ ; λ = 3, 4
⎡ ⎤ 1 ⎢ ⎥ Λ = ⎣2⎦ mod 7 3
(5.45)
We note that the two new eigenvalues are nondegenerate. Each belongs to the third row of one of the newfound eigenvectors, because they were determined from the eigenvalue equation for H (3) . This set of unique eigenvectors simultaneously diagonalizes all the class-coefficient matrices. We note that the second row of both of the newfound eigenvectors contains the class constants λ = 2, which were doubly degenerate eigenvalues for the eigenvalue equation for H (2) . Here they are components of the eigenvectors, but not eigenvalues themselves.
128
Dixon’s method for computing group characters Exercises
5.1 Assume that λ is an eigenvalue for the matrix operator a b M= . c d Carry out row operations in algebraic form to demonstrate explicitly that the augmented matrix provides the eigenvector belonging to λ. 5.2 Show that the eigenvectors obtained in (5.44) are linear combinations of the starting basis vectors of (5.43).
Appendix 2 In Appendix 1, the solution to the eigenvalue equation H (i)(α) Λ = (α) λi
(α)
Λ
(5.46)
was described. In the example given there, the eigenvalues mod 7 were found to be λ = 6, 4, 3 with corresponding eigenvectors ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (5.47) ⎣2⎦ mod 7. ⎣2⎦, ⎣6⎦, 3 4 0 Each eigenvector corresponds to an Irrep, and each component of the eigenvector is a class constant λ for one of the classes in that Irrep. A computer program must keep track of the classes and number them. Since the classconstant matrix H (1) is always diagonal, the computer program will begin with H (2) . Of course, what is called H (2) depends on the generators chosen and the order in which they are entered into the program. In our example case, taken from Appendix 1, H (2) corresponds to the class of rotation operators {C3 , C32 }. A different choice of generators, or order of entry, could have caused H (2) to correspond to the class of reflection operators {σ1 , σ2 , σ3 }. One must be sure to keep them all straight. We can determine the character for the elements of each class by using (5.28), which we rewrite here as dα Θ (α) χi ≡ (5.48) Θ (α) λi mod p, nc(i) where Θ denotes a modular character. The number of elements, nc(i), in each class, Ci , is known. Only the dimension of the Irrep, dα , is needed to determine the modular character Θ((α) χi ). We can find dα using (5.29), which is g mod p, (5.49) |dα |2 ≡ Θ((α) λi )Θ((α) λ−1 ) i nc(i) i where (α) λ−1 means the class constant for the inverse class mod p. That is, (α) λi is the i class constant of the ith class. One takes an element of the ith class and finds its inverse. The inverse belongs to some class and (α) λ−1 means the class constant for that inverse i
Appendix 2
129
class. However, even when we find Θ((α) χi ), it will be modulo a prime number, in this case p = 7, and there is a bit more modular arithmetic involved before we get to the point of finding the multiplicities that allow us to express the character as a polynomial in roots of unity. We will work through an example using our example group C3v . Dixon’s method Remembering that Dixon’s method uses modular arithmetic, we evaluate (5.49) for the eigenvalue (the class constant) λ = 6 and the eigenvector [1 6 0], which corresponds to the Irrep to which λ = 6 belongs. |dα |2 ≡
6 mod 7, 6×6 0×0 1×1 + + 1 2 3
(5.50)
where 1, 2, and 3 are, respectively, the numbers of elements in the classes (the orders of the classes), the eigenvector is the first of the three eigenvectors in (5.47), and the modulus 7 was determined in Appendix 1. In general, the orders of classes, the modulus, and anything else needed at this point would be included in the computer program output. The equation for dα can be rewritten 6 mod 7 1 + 18 + 0 6 ≡ mod 7 5
|dα |2 ≡
(5.51)
There are numerous ways to determine dα such that |dα |2 < 7 with dα > 0.
Example 5.7 6 6 3 18 14 + 4 4 ≡ × ≡ ≡ ≡ ≡ 4 mod 7 5 5 3 15 14 + 1 1 ⇒ d2j = 4 mod 7.
d2 ≡
or d2 ≡
6 6 6 ≡ ≡ ≡ −3 ≡ 4 mod 7. 5 5−7 −2
But what is needed for computer programs is a systematic method. Fermat’s theorem provides it.
Fermat’s theorem In the 1600s, Fermat discovered a beautiful and useful theorem: Theorem: If p is a prime number that does not divide the integer a, then ap−1 ≡ 1 mod p.
(5.52)
130
Dixon’s method for computing group characters
In general, we have an expression of the form a d2 ≡ mod p. b
(5.53)
We convert the denominator to unity with the aid of Fermat’s theorem. This gives d2 ≡
a abp−2 bp−2 × p−2 ≡ p−1 ≡ abp−2 mod p. b b b
(5.54)
We can do all this in a Mathematica program by making use of the function Mod[i,j], which with an integer argument gives the remainder of i divided by j. Thus we have d2 ≡ abp−2 mod p = Mod[abp−2 , p].
(5.55)
We evaluate fractions at each step by this procedure. Thus, in (5.50), we evaluate each fraction in the denominator using (5.55), and then we add all the terms in the denominator, again using modular arithmetic. We continue in this fashion, using modular arithmetic at each step, and not waiting until the end to use it.
Modular characters Using Fermat’s theorem to solve for |dα |2 we find that |dα |2 ≡ 4 (mod 7). There is an obvious solution, namely, dα = 2. But there are two roots, and the other one is equivalent to −2, or 5. In general, the roots of d2 (mod p) are d, p − d. In writing a computer program, one must be sure to find both roots, subject to the requirement that p > 2dα , and then choose the smaller root to ensure that dα is uniquely and correctly determined.
Example 5.8 Suppose we have calculated |dα |2 ≡ 16 (mod p), and we know the correct answer to be 4. What do we get for dα using several different prime numbers for p? dα ≡ 4, 1 mod 5, dα ≡ 4, 3 mod 7, dα ≡ 4, 7 mod 11, dα ≡ 4, 13 mod 17. We always get the correct root along with another that may be more or less than the correct one. By choosing p > 2dα we ensure that the smaller modular root is the correct one.
Appendix 2
131
√ More generally, we take p > 2 g to ensure that we uniquely determine dα for all Irreps. Proof Let D equal the dimension of the largest Irrep. This Irrep appears D times in the regular representation, which is of dimension g. Thus √ D2 ≤ g, D ≤ g. By initially choosing
√ p > 2 g,
we ensure that p > 2dα for any Irrep. With a value of prime number p sufficiently large, but not larger than necessary, one can just go methodically through the squares of integers, beginning with 1, to find the first value of n such that n2 = d2 . Since the two roots are symmetrical about p/2 one need consider only squares of n with n ranging from 1 to (p − 1)/2. Now we are prepared to solve for the irreducible modular characters using (5.48). Working through the case for the first of the eigenvectors in (5.47), we find Θ(χ1 ) =
2 × 1 = 2, 1
Θ(χ2 ) =
2 × 6 = 6, 2
2 × 0 = 0. 3 We can now use these irreducible modular characters to calculate the multiplicities in the polynomial expression for the character. Θ(χ3 ) =
Computing multiplicities (α)
Now that the characters Θ( χi ) are known, we can compute the multiplicities from (5.30). Here, we drop the Irrep designation (α) for simplicity and write mi (s) =
n−1 1 Θ(χi (At ))Z −st (mod p), n t=0
(α)
mi (s)
(5.56)
where At is a power of any element A belonging to the class Ci . We begin with the trivial case of finding the multiplicities for the first class, the identity class, for the Irrep of dimension 2, which corresponds to the eigenvector [1 6 0]. Since all powers of the identity element are equal to the identity element, the irreducible modular characters Θ(χ1 (At )) = Θ(χ1 (E t )) = Θ(χ1 (E)) are all the same, namely, they all equal 2, as was found in the previous section. Thus, 1 × 2 × (Z 0 + Z −s + Z −2s + Z −3s + Z −4s + Z −5s ). n For our example case, n = 6 and Z = 3, thus, m1 (s) =
m1 (0) =
1 1 × 2 × (30 + 30 + 30 + 30 + 30 + 30 ) = × 2 × 6 = 2, 6 6
(5.57)
Dixon’s method for computing group characters
132
which is just what we expect for a two-dimensional Irrep for the identity element. Just as a check, let us consider another multiplicity: 1 0 (3 + 3−1 + 3−2 + 3−3 + 3−4 + 3−5 ) 3 1 1 1 1 1 1 1+ + + + + ≡ 3 3 9 27 81 243 1 1 1 1 1 1 ≡ 1+ + + + + (mod 7) 3 3 2 6 4 5
m1 (1) ≡
≡ 0.
(5.58)
Similarly, all other values of m1 (s), for s > 0, are found to equal zero. Now let us consider the class C3 , for which the irreducible modular character is zero. At first glance, one might think that the multiplicities would also be zero. But note that one sums over the modular characters of powers of some element belonging to the class. An element raised to some power often belongs to a different class with a nonzero character. Our example physical realization of the class C3 consists of the three reflection operations of the equilateral triangle. We can use σ1 as a representative operation. Odd powers of σ1 are equal to σ1 itself, but even powers are equal to the identity element E for which the irreducible modular character is 2. Thus, the modular characters in (5.56) will alternately be 2 and 0, giving 1 (2 × 30 + 0 × 3−s + 2 × 3−2s + 0 × 3−3s + 2 × 3−4s + 0 × 3−5s ), 6 1 (5.59) ≡ (30 + 3−2s + 3−4s ), mod 7. 3
m3 (s) ≡
We can now determine the values of m3 (s): 1 0 (3 + 30 + 30 ) ≡ 3 1 m3 (1) ≡ (30 + 3−2 + 3−4 ) 3 1 1 1 ≡ 1+ + ≡ 3 2 4
m3 (0) ≡
1 × 3 ≡ 1. 3 1 1 1 ≡ 1+ + 3 9 81 1 (1 + 4 + 2) 3
≡ 0.
1 0 1 1 1 (3 + 3−4 + 3−8 ) ≡ 1+ + 3 3 81 6561 1 1 1 1 ≡ 1+ + ≡ (1 + 2 + 4) 3 4 2 3
m3 (2) ≡
≡ 0.
1 0 1 1 1 (3 + 3−6 + 3−12 ) ≡ 1+ + 3 3 729 531441 1 1 1 1 ≡ 1+ + ≡ (1 + 1 + 1) 3 1 1 3
m3 (3) ≡
≡ 1.
Computation project
133
The remaining values are easily found to be m2 (4) = 0,
m2 (5) = 0.
Of course, in a computer program, or even when doing such a calculation by hand, one would always shift numbers at each stage to be integers in the range 0 to p − 1. Here, large integers mod 7 have been kept at intermediate stages just to make clear the steps in the calculation. Following the procedures just illustrated, we find the multiplicities for the class C2 to be m2 (0) = m2 (1) = m2 (3) = m2 (5) = 0. m2 (2) = m2 (4) = 1.
Irreducible characters The multiplicities are now known for expressing the irreducible characters as a polynomial in sixth-roots of unity. We have for the two-dimensional Irrep: χ(C1 ) = m1 (0)e0 = 2. χ(C2 ) = m2 (2)ei(2π(2/6)) + m2 (4)ei(2π(4/6)) = ei(2π/3) + ei(4π/3) √ √ 1 1 3 3 = − +i + − −i 2 2 2 2 = −1. χ(C3 ) = m3 (0)e0 + m3 (3)ei(2π(3/6)) = e0 + eiπ =1−1 = 0.
Computation project Use the sample programs given in this chapter together with other sample programs presented in Chapters 2 and 4 to construct a complete program to determine the characters of finite groups. Use the program you develop to determine the Irreps of the point-groups C4v and C6v .
6 Group action and symmetry projection operators
In the first part of this chapter we introduce some concepts regarding the action of symmetry group operators on objects, sets, and number fields; and we define the terminology involved. This is followed in the second part by the definition of symmetry projection operators, and description of the methodology for their construction. We also demonstrate how the emerging ideas of group action enter into the operational structure of the projection operators.
6.1 Group action We would like to stress the fact that the application of group theory to a physical system entails defining the changes effected by the operators of its symmetry group on its constituent elements. In other words, describing the group action on the system. So far, we have presented ample, but particular, examples of this action, such as geometric rotations of, and permutations among the system constituents. However, it is now desirable to present some concepts in group actions that transcend the particulars we have considered previously. These concepts are particularly evident in geometric situations; we find such examples in crystallography, where basic equivalence relations naturally emerge, and become algorithmically approachable. Let us examine the case of a group G acting on a set of points M ; we then note the following: (i) G-set The action of G on M is a mapping where every element g ∈ G takes a point m ∈ M to the point m = gm ∈ M such that g2 (g1 m) = (g2 g1 ) m, ∀g1 , g2 ∈ G, ∀m ∈ M. (6.1) E m = m, We note that on the left-hand side the action map is applied twice, once for g1 and then for g2 , while on the right-hand side it is applied only once, for the group product g1 g2 = g3 . The action of G on M qualifies M as a G-set. (ii) Orbit When any element g ∈ G acts on M , such that g m1 = m2 , the two elements m1 , m2 ∈ M are said to lie in the same orbit. Thus, a G-orbit OG (m) is the subset of + , points g m ∀g ∈ G . 134
6.1 Group action
135
It is then clear that different orbits generated by the action of G on M are either identical or disjoint subsets of M . The G-orbits, + , OG (m) := gm g ∈ G ∀ m ∈ M partition M into pairwise disjoint subsets.
Example 6.1 The action of C2v on the water molecule We find from Figure 6.1 that when we apply the operations of C2v to the H2 O molecule, i.e. to the set [H1 , H2 , O], O is left invariant, while H1 and H2 are interchanged, or permuted, under the action of the two-fold rotation C2 and the reflection σy , but they are left unchanged by σx and, of course, E. Therefore, the action of C2v results in two C2v -orbits: – A two-member orbit OG (H1 ) = [H1 , H2 ], and – A one-member orbit OG (O) = [O]. z C2 O
sx H1 H2
sy
y x
Fig. 6.1. The water molecule and its symmetry.
Example 6.2 Orbits under the action of C4v on a square sheet Typical orbits generated by the action of C4v on the continuous set of points forming a square are shown in Figure 6.2; we find four types of orbits: namely, one 8-member orbit, two 4-member orbits, and one 1-member orbit.
136
Group action and symmetry projection operators
8-member orbit 4-member orbit 4-member orbit
1-member orbit
Fig. 6.2. Some orbits of the action of C4v on a square.
Being in the same orbit is an equivalence relation on M, sometimes denoted by ∼G . The set M/ ∼G of equivalence classes or orbits is usually denoted by GM (and called the quotient M mod G). In mathematics, an orbit is a very general group-theoretical term describing any set of objects that are mapped onto each other by the action of a group. However, we will present in Chapters 10–12 more tangible manifestations of orbits associated with crystallographic symmetry and the Irreps of space-groups: • An orbit can be identified with a lattice because it is generated by the action of its space-group on any single point in a Euclidean space. • Another orbit is identified as a Wyckoff position or set when the action is due to a crystallographic point-group. • In the classification of space-group Irreps, Chapter 11, an orbit is manifest in the star of the wavevector. (iii) Stabilizer Sometimes the action of some elements of G on m ∈ M leaves m unchanged. We then identify the set of these operators as the stabilizer of the element m in G, Gm := {g ∈ G|gm = m}. Clearly, Gm is a subgroup of G of order gm , namely, Gm ⊆ G. The orbit to which m belongs contains g/gm members; for each member we identify a stabilizer subgroup of G of order gm . The collection of stabilizers of these members constitute conjugacy subgroups of Gm in G, as will be explained in Chapter 8. The intersection of all conjugacy classes of stabilizers Gm with m ∈ M is called the kernel of the action, and is a normal subgroup of G. Sometimes the stabilizer is referred to as an isotropy group; however, we reserve this label to describe subgroups associated with the Landau theory of phase transitions, presented in Chapter 17.
6.1 Group action
137
Example 6.3 The stabilizers associated with the orbit points of Example 6.2
8-member orbit
→ {E}
4-member orbit
→ {E, σv } → {E, σx }, {E, σy } (conjugate subgroups),
4-member orbit
→ {E, σd } → {E, σxy }, {E, σx¯y } (conjugate subgroups),
1-member orbit
→ C4v .
• The 1-member orbit, with C4v as a stabilizer, is called a fixed-point orbit. • The 8-member orbit, with {E} as a stabilizer, is called a principal orbit.
In different applications of symmetry discussed in later chapters, we will encounter manifestations of stabilizers under different guises; for example, in crystallographic applications we identify stabilizers under the name site-symmetry groups in real space; while for wavevectors in reciprocal space they are known as the little groups of the wavevectors. (iv) Orbit type Orbits with the same conjugacy classes of stabilizers are said to be of the same type.
Example 6.4 The square center (open circle) is a fixed-point orbit, it is the only such point on the square sheet. We also find that the two 4-member orbits cited above are not of the same type, since they belong to the distinct conjugacy classes {E, σv } and {E, σd }.
(v) Stratum action.
A stratum is the union of orbits of the same type that result from a group
Example 6.5 Square-sheet strata We identify the strata, starting from top left, as a fixed-point origin, a {E, σv }-type stratum, and a {E, σd }-type stratum. The remaining orbits are of the principal type and form a single stratum.
138
Group action and symmetry projection operators
1-member stratum
4-member stratum
4-member stratum
8-member stratum
Fig. 6.3. Strata of the action of C4v on a square sheet.
6.2 Symmetry projection operators Given a vector in ordinary three-dimensional space, it is sometimes necessary to know its projection onto a plane or onto another vector. These ideas are familiar from analytic geometry. They are generalized in quantum mechanics where the ordinary vector becomes a state vector, or state function |Ψ, using the ket-bra notation, in Hilbert space. For example, if a set of orthonormal functions {|φn } forms a complete basis for a Hilbert space, it is possible to perform a decomposition, or projection, of a state function |Ψ onto the constituent basis functions by employing the set of projection operators Pˆn :
Pˆn |Ψ = |φn φn Ψ = cn |φn ,
(6.2)
so that |Ψ =
n
|φn φn Ψ = cn |φn .
(6.3)
n
The projection operator can be viewed as a generalization of the idea of resolving a vector into its components along the basis vectors of a Cartesian coordinate system. In group theory it resolves a state vector, a function, into components that transform according to the rows of the different Irreps of the symmetry group. In contrast to quantum mechanical projection operators, in Hilbert spaces, and in vector projections in Cartesian spaces, this projection is independent of the particular basis set of functions that may be associated with the Irreps. These symmetry projection operators are constructed from the matrix representatives of Irreps.
6.2 Symmetry projection operators
139
6.2.1 Construction of the symmetry transfer operators + , Let us consider a basis set (ν)φk for an Irrep (ν) of a group G. If we operate on one of ˆ we generate a linear combination of the the basis functions, (ν)φk , with the operator R, set of basis functions given by (ν) (ν) ˆ (ν)φk = φl Γlk (R), (6.4) R l
where the coefficients are elements of the kth row of orthogonality relations given by (4.27), dµ g R
(µ)
(ν)
Γ(R). Making use of the matrix
Γij (R) (ν) Γ−1 j i (R) = δµν δii δjj ,
(6.5)
we multiply (6.5) by (dµ /g) (µ) Γ−1 ji (R) and sum over R, to get dµ (µ) −1 ˆ Γji (R) R g R
(ν)
φk =
(ν)
φl
l
=
(ν)
dµ g R
(µ)
(ν) Γ−1 Γlk (R) ji (R)
φl δνµ δil δjk
l
=
(ν)
φi δνµ δjk .
(6.6)
It is now possible to define a symmetry transfer operator
(µ)
dµ Pˆij = g
(µ)
(µ)
ˆ Γ−1 ji (R) R,
Pˆij as
i = j.
(6.7)
ˆ R
The matrix element (µ) Γ−1 ji (R) is just a number, perhaps a complex number. The operator ˆ ˆ on an arbitrary complete set R that follows can be expressed in any Rep engendered by R of basis functions in the Hilbert space of the physical system invariant under G. The nature ˆ will be demonstrated in the examples and procedure of construction of the operator R that follow. We find that (µ)
Pˆij† (ν) Pˆi j = δµν δii
(µ)
Pˆjj .
(6.8)
The restriction i = j in (6.7) is a reminder that (µ) Pˆij is an off-diagonal operator. It is a symmetry transfer operator which changes the symmetry properties from one row of the Irrep (ν) to another row of the same Irrep, as described by the equation
(µ)
Pˆij
(ν)
φk =(ν) φi δνµ δjk .
(6.9)
Group action and symmetry projection operators
140
If ν = µ, k = j, then (µ)
Pˆij
(µ)
φj =(µ) φi .
(6.10)
That is, the symmetry transfer operator Pˆij operating on a function φj belonging to the jth row of the µth Irrep yields its orthogonal partner belonging to the ith row. Thus, if one of the basis functions is known we can find all its partners by using the symmetry transfer operator. An example follows. (µ)
(µ)
Example 6.6 We treat a system with C3v symmetry, and consider the function, φ2 = x2 − y 2 , which transforms according to the second row of its two-dimensional Irrep (3) Γ. Operating with the symmetry transfer operator 2 (3) −1 (3) ˆ ˆ Γ21 (R) R P21 = 6 R 1 (3) −1 ˆ + (3) Γ−1 Γ21 (E) E ˆ1 + (3) Γ−1 ˆ2 = 21 (σ1 ) σ 21 (σ2 ) σ 3 . (3) −1 ˆ ˆ32 + (3) Γ−1 ˆ3 + (3) Γ−1 Γ21 (C32 ) C 21 (C3 ) C3 + 21 (σ3 ) σ √ √ 1 3 ˆ2 3 ˆ ˆ+σ ˆ2 ) + ˆ1 ) = (C3 + σ (C3 + σ 0 (E ˆ1 ) − 2 2 3 should produce its partner function, which belongs to the first row of (3) Γ. ˆ3 on φ2 . Following We remind ourselves of the procedure by considering the action of C the prescription of Section 3.2, we replace x and y in φ2 (x, y) = x2 − y 2 by the corresponding components x and y as determined by √ −1 − 23 x x −1 x 2 = C3 = √ . 3 1 y y y − 2
2
We find after a bit of algebra that −1 2 ˆ3 (x2 − y 2 ) = σ ˆ3 (x2 − y 2 ) = x + C 2 −1 2 ˆ32 (x2 − y 2 ) = σ ˆ2 (x2 − y 2 ) = C x + 2 We obtain (3)
1 Pˆ12 (x2 − y 2 ) = 3
1 2 √ y + 3xy, 2 1 2 √ y − 3xy. 2
-
2 × 0 × (x2 − y 2 ) √ 3 −1 2 1 2 √ −2× x + y − 3xy 2 2 2 √ . 3 −1 2 1 2 √ +2× x + y + 3xy 2 2 2
= 2xy. Thus 2xy is the partner function of x2 − y 2 and the two form a basis for
(3)
Γ.
6.2 Symmetry projection operators
141
In general, if we know just one basis function for an Irrep, we can find all the partner basis functions by using the appropriate symmetry transfer operators.
ˆ Construction of the operator R ˆ when applying At this point we try to clarify the physical role played by the operator R (6.7) to a physical problem. In a typical problem we may be interested in the entire Hilbert space of the system, or in some closed subspace. In any case, we can choose a basis set that spans the relevant space. We partition the relevant space into subspaces that belong to different rows, indexed k, of Irreps (ν). We also allow for the possible multiple occurrence of subspaces belonging to Irrep (ν), and label these (ν)-subspaces with the index s. Thus + , we designate the basis as Φ := (ν)φks . A familiar example of such multiple occurrence can be found, say, in the p-states of a hydrogen atom; they may be 2p, 3p, 4p, . . . but they all transform like the basis Y1m ! Now, we expand an arbitrary state vector Ψ in the relevant Hilbert space/subspace of the physical system, in terms of the basis set {(ν)φks } that span the Hilbert space. We obtain (ν) Ψ= cks (ν) φks , (6.11) ν,k,s
where k and s index the rows of Irrep (ν) and the subspaces that transform according to Irrep (ν). Operating on Ψ by the symmetry transfer operator (µ) Pˆij gives (µ)
Pˆij Ψ =(µ) Pˆij =
(ν)
cks
(ν)
ν,k,s (ν)
cks
φks =
(ν)
cks
(µ)
Pˆij
(µ)
φis ,
(ν)
φks
ν,k,s (ν)
φis δνµ δjk =
(µ)
cjs
(6.12)
s
ν,k,s
where we used (6.9). We find that (µ) φis s.
(µ)
Pˆij selects the
(µ)
φjs s from Ψ and converts them to
6.2.2 The Wigner projection operator By setting j = i, we can define a projection operator known as the Wigner projection operator (µ) Pii as (µ)
Pˆii Ψ =(µ) ci
(µ)
φi .
(6.13)
It is of fundamental importance. In addition to its many applications in physical problems, it will play a key role in developing the irreducible matrix Reps. We identify (µ) Pˆii as the projection operator associated with the ith row of the µth Irrep, the (µ, i)th symmetry of G, or the Wigner projection operator. The projection operator selects the component of an arbitrary vector Ψ along (µ) φi , which may be thought of as
Group action and symmetry projection operators
142
the component along a unit vector (µ) φi inasmuch as (µ) φi is from an orthonormal basis set. It is clear that when the projection operator is applied to a general function, it will yield either zero or a basis function that is unnormalized, in general.
Example 6.7 We consider the operation of the Wigner projection operators belonging to the (3) Γ of C3v , acting on: (i) The function F(x, y, z) = x + y + z + xz + x2 . Here, we must do something a bit different from Example 6.1. In that example we operated on a known basis function belonging to one row of an Irrep, and converted it to a basis function belonging to a different row of the Irrep. Here, we are considering a rather general function, or Hilbert space vector, and we want to project out basis functions that belong to a particular row of an Irrep. One way of doing the projection is to follow a procedure analogous to what was done in Section 3.2. The prescription To project (µ) Pˆii onto F(x, y, z), write the parts of the function F as the components of a vector and let the projection operator (µ)
dµ Pˆii = g
(µ)
ˆ ˆ Γ−1 ii (R) R
(6.14)
ˆ R
act on each component separately. This generates a new function F , some (or all) of whose components might be zero. The components of this new function are basis functions that belong to the Irrep (µ) Γ. For (3) Γ of C3v , we take i = 1 and refer to Section 3.2 for the action of individual ˆ on the various parts of the function F. By choosing i = 1 we find basis operators R functions that belong to the first row of (3) Γ. ⎡ d3 (3) −1 ˆ ˆ (3) ˆ Γ11 (R) R P11 F = g ˆ R
x
⎤
⎢ ⎥ ⎢y⎥ ⎢ ⎥ ⎢z⎥ ⎢ ⎥ ⎢ ⎥ ⎣xz ⎦ x2
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
2 ˆ = (1) E 6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
⎡
x
⎤
⎢ ⎥ ⎢y⎥ ⎢ ⎥ ⎢ z ⎥ + (−1) σ ˆ1 ⎢ ⎥ ⎢ ⎥ ⎣xz ⎦ x2
⎡
x
⎤
⎡
x
⎤
⎢ ⎥ ⎢ ⎥ ⎢y⎥ ⎢y⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎢z⎥ + ⎢ ⎥ σ ˆ 2 ⎢ z ⎥ ⎢ ⎥ 2 ⎢ ⎥ ⎢ ⎥ ⎣xz ⎦ ⎣xz ⎦ x2
x2
6.2 Symmetry projection operators
143
⎡
⎤ ⎡ ⎤ x x ⎢y⎥ ⎢y⎥ ⎢ ⎥ ⎢ ⎥ 1 −1 ˆ ⎢ ⎥ −1 ˆ 2 ⎢ ⎥ + σ ˆ3 ⎢ z ⎥ + C3 ⎢ z ⎥ + C3 ⎢ ⎥ ⎢ ⎥ 2 2 2 ⎣xz ⎦ ⎣xz ⎦ x2 x2 ⎧⎡ ⎤ ⎡ ⎤ −x ⎪ x ⎪ ⎪ ⎥ ⎪ ⎢ ⎥ ⎢ ⎪ ⎨⎢ y ⎥ ⎢ y ⎥ 1 ⎥ ⎢ ⎥ ⎢ = ⎢z ⎥−⎢ z ⎥ ⎥ ⎢ ⎥ ⎢ 3 ⎪ ⎪ ⎪ ⎣xz ⎦ ⎣−xz ⎦ ⎪ ⎪ ⎩ 2 x2 x ⎡
√ 1 x − 23 y 2√ − 23 x − 12 y
⎤
⎡
√ 1 x + 23 y 2√ + 23 x − 12 y
⎤⎫ x ⎪ ⎪ ⎢ y ⎥⎪ ⎪ ⎢ ⎥⎪ ⎬ ⎢ ⎥ ⎢z⎥ ⎢ ⎥⎪ ⎪ ⎣xz ⎦ ⎪ ⎪ ⎪ 2 ⎭ x ⎡
⎤
⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ 1 ⎢ 1 ⎢ ⎥ ⎥ z z + + ⎥ ⎥ ⎢ ⎢ √ √ 2 ⎢ 1 x − 3 y z⎥ 2 ⎢ 1 x + 3 y z⎥ ⎥ ⎥ ⎢ 2 ⎢ 2 2 ⎣ ⎣ 2 ⎦ 2 ⎦ √ √ 1 3 1 3 x− 2 y x+ 2 y 2 2 √ ⎤⎫ 1 ⎪ x + 23 y −√ ⎪ 2 ⎥ ⎥⎪ ⎪ ⎢ ⎢ − 3 1 x − 2y ⎥⎪ ⎥ ⎪ ⎢ ⎢ 2 ⎥ ⎥⎪ ⎬ ⎢ ⎢ 1 ⎢ ⎥ 1 ⎢ ⎥ z z − ⎢ ⎥ − ⎢ ⎥ √ √ 2 ⎢ −1 x − 3 y z ⎥ 2 ⎢ −1 x + 3 y z ⎥ ⎪ ⎥ ⎥⎪ ⎪ ⎢ 2 ⎢ 2 2 2 ⎪ ⎣ ⎣ 2 ⎦ 2 ⎦ ⎪ √ √ ⎪ ⎪ −1 −1 ⎭ x − 23 y x + 23 y 2 2
⎡
√ −1 x − 23 y 2√ + 23 x − 12 y
⎤
⎡
⎤ ⎡ ⎤ x 3x ⎢ 0 ⎥ ⎢0⎥ ⎥ ⎢ ⎥ ⎢ 1 ⎢ ⎥ ⎢ ⎥ = ⎢ 0 ⎥ = ⎢ 0 ⎥. ⎥ ⎢ ⎥ 3 ⎢ ⎣3xz ⎦ ⎣xz ⎦ 0 0 ⎡
We have found two basis functions {x, xz} that belong to the first row of (3) Γ for the group C3v . We might expect from Section 3.2 that x2 would be a basis function. Perhaps it belongs to the second row of (3) Γ. (ii) The function, F = x2 + y 2 + xy, and check to see if any parts of this function serve as basis functions belonging to the second row of (3) Γ.
dµ (µ) ˆ P22 F = g
(µ)
Γ−1 22
ˆ R
⎧ ⎪ ⎨
⎡ ⎤ x2 ⎢ 2⎥ ˆ ˆ R R ⎣y ⎦ xy
⎤ ⎡ ⎤ ⎡ ⎤ x2 x2 x2 2 −1 ⎢ 2⎥ ⎢ 2⎥ ⎢ 2⎥ ˆ = ˆ1 ⎣ y ⎦ + (1) E ⎣ y ⎦ + (1) σ σ ˆ2 ⎣ y ⎦ 6 ⎪ 2 ⎩ xy xy xy ⎡
Group action and symmetry projection operators
144
+
⎤ ⎡ ⎤ ⎡ ⎤⎫ x2 x2 x2 ⎪ ⎬ −1 −1 ⎢ 2⎥ ⎢ 2⎥ 2 ⎢ 2⎥ ˆ ˆ C3 ⎣ y ⎦ + C3 ⎣ y ⎦ ⎣y ⎦ + ⎪ 2 2 xy xy xy ⎭ ⎡
−1 2
σ ˆ3
⎧ ⎡ ⎤ ⎡ ⎤ ⎪ ⎪ 2 ⎪ x2 ⎨ x −1 1 ⎢ 2 ⎥ ⎢ 2⎥ = ⎣y ⎦ + ⎣ y ⎦ + ⎪ 3 ⎪ 2 ⎪ −xy ⎩ xy ⎡ −
⎢ 1 ⎢ ⎢ 2 ⎣
⎢ 1 ⎢ ⎢ 2 ⎣
⎤ 2 √ 1 x − 23 y 2 ⎥ 2 √ ⎥ − 3 ⎥ x − 21 y 2 ⎦ √ √ − 3 − 23 y x − 12 y 2
⎢ ⎢ ⎢ ⎣
1 x 2
⎤ 2 √ 1 x + 23 y 2 ⎥ 2 √ ⎥ 3 ⎥ x − 21 y 2 ⎦ √ √ 3 + 23 y x − 12 y 2
⎡
1 x 2
⎡ −
⎡
−
⎢ 1 ⎢ ⎢ 2 ⎣
⎤ 2 √ −1 x − 23 y 2 ⎥ 2 √ ⎥ 3 ⎥ x − 12 y 2 ⎦ √ √ 3 − 23 y x − 21 y 2
−1 x 2
⎤⎫ 2 √ −1 ⎪ x + 23 y ⎪ 2 ⎥⎪ ⎬ 2 √ ⎥ − 3 ⎥ x − 12 y 2 √ ⎦ ⎪ ⎪ √ ⎪ − 3 ⎭ + 23 y x − 12 y 2
−1 x 2
⎤ ⎡ ⎤ ⎡ 3 (x2 − y 2 ) (x2 − y 2 ) 1 ⎢ 23 2 ⎥ ⎢ 2 2 2 ⎥ = ⎣ 2 (y − x ) ⎦ = ⎣(y − x )⎦, 3 0 0 where a factor of 1/2 has been dropped in the last matrix in order to get the conventional normalization. Clearly, any numerical multiplier will be carried through all steps of the projection process, so the projected basis functions are not generally normalized. We have projected out the basis function (x2 −y 2 ) as belonging to the second row of (3) Γ for the group C3v . In Chapter 3, we recognized that it was necessary to take the appropriate linear combination of x2 and y 2 to get a basis function belonging to a unitary Rep. Here, the use of unitary projection operators automatically gave us the correct basis function, apart from a scaling factor.
Hermiticity and idempotency: the eigenvalue problem The Wigner projection operators are Hermitian and idempotent, namely (µ)
Pˆii† =
(µ) ˆ Pii ,
(6.15)
(µ)
Pˆii2 =
(µ) ˆ Pii .
(6.16)
Let us introduce an arbitrary complete set of basis vectors in the Hilbert space of the system , + Ψ = φ1 , φ2 , . . . , which engenders a Rep of G ˆ Ψ = Ψ Γ(R), R
R ∈ G.
6.2 Symmetry projection operators " in the expression of (µ) Pii , and we obtain Γ(R) is then substituted for R dµ (µ) ∗ (µ) ˆ Γii (R) Γ(R) = Ψ (µ) Pi . Pii Ψ = Ψ g R∈G The matrix
(µ)
145
(6.17)
Pi satisfies [5]:
(i) (6.15), which guarantees the existence of a unitary transformation U that diagonalizes (µ) Pi , and (ii) (6.16), which tells us that the eigenvalues can only be 1 or 0 !! The eigenvalue problem We then write
U†
(µ)
Pi U =
or
(µ)
1 0
Pi U = U
1 0
0 0
(6.18)
0 = u, 0 , 0
(6.19)
where u is the rectangular part of U containing eigencolumns with unit eigenvalues, (µ)
Pi u = u.
(6.20)
Adopting this approach ensures that the ensuing symmetrized vector, i.e. the eigencolumns of u, are independent.
Example 6.8 We consider the case of vibrational modes of the water molecule. The H2 O molecule has C2v symmetry, with the symmetry operations shown in Figure 6.4. z C2 O
sx H1 sy
H2 y
x
Fig. 6.4. The water molecule and its symmetry.
146
Group action and symmetry projection operators
Hilbert space The relevant Hilbert space is that of infinitesimal displacements of the O and two H atoms, of dimension 9; it is spanned by the vector # d = dH1 # = xH1
$ dH2
dO
yH1
zH1
xH2
yH2
zH2
xO
yO
$ zO ,
where, for example, dH1 = [xH1 yH1 zH1 ] describes the displacements of the hydrogen atom 1, as shown in Figure 6.4. Rep engendered by C2v We find that the local displacements of each atom, di = [xi yi zi ], will transform, under the operations of C2v , according to the matrices: ⎡
1 0 ⎢ I di = ⎣0 1 0 0 ⎡ −1 0 ⎢ σx d i = ⎣ 0 1 0 0
⎤ ⎤ ⎡ 0 −1 0 0 ⎥ ⎥ ⎢ C2 d i = ⎣ 0 0⎦ di , −1 0⎦ di , 1 0 0 1 ⎤ ⎤ ⎡ 0 1 0 0 ⎥ ⎥ ⎢ 0⎦ di , σy di = ⎣0 −1 0⎦ di . 1 0 0 1
In addition, we should remember that the action of C2v on the H2 O molecule induced permutations among the set [H1 , H2 , O], which can be described by the matrices ⎞ ⎡ 0 H1 ⎜ ⎟ ⎢ C2 ⎝H2 ⎠ = ⎣1 0 O ⎛ ⎞ ⎡ 1 H1 ⎜ ⎟ ⎢ σx ⎝H2 ⎠ = ⎣0 0 O ⎛ ⎞ ⎡ 0 H1 ⎜ ⎟ ⎢ σy ⎝H2 ⎠ = ⎣1 0 O ⎛
1 0 0 0 1 0 1 0 0
⎤ 0 ⎥ 0⎦ 1 ⎤ 0 ⎥ 0⎦ 1 ⎤ 0 ⎥ 0⎦ 1
⎛
⎞ H1 ⎜ ⎟ ⎝H2 ⎠, O ⎛ ⎞ H1 ⎜ ⎟ ⎝H2 ⎠, O ⎛ ⎞ H1 ⎜ ⎟ ⎝H2 ⎠. O
Combining these into a single Rep that includes the effect of symmetry operations of C2v on both the atoms and the local coordinate systems attached to the atoms, we obtain ⎤ ⎡ ⎤ I 0 0 0 C2 0 ⎥ ⎢ ⎢ ⎥ I = ⎣0 I 0⎦, C2 = ⎣C2 0 0 ⎦, 0 0 I 0 0 C2 ⎡ ⎡ ⎤ ⎤ σx 0 0 σy 0 0 ⎢ ⎢ ⎥ ⎥ σx = ⎣ 0 σx 0 ⎦, σy = ⎣σy 0 0 ⎦, 0 0 σx 0 0 σy ⎡
where C2 , σx and σy are the 3 × 3 matrices define above, and 0 is the 3 × 3 null matrix.
6.2 Symmetry projection operators
147
The Irreps of C2v can be simply determined by the inspection method, and are given in Table 6.1. Table 6.1. Character table for C2v . E
C2
σx
σy
(1)
Γ
1
1
1
1
(2)
Γ
1
1
−1
−1
(3)
Γ
1
−1
1
−1
(4)
Γ
1
−1
−1
1
As an example, let us construct the projection operator
(1)
1 [I 4 ⎡ I ⎢ = ⎣0 0
P11 =
0 ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢0 ⎢ = ⎢ ⎢0 ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎣0 0
Diagonalization of
P11 using (6.14),
+ C2 + σx + σy ] 0 I 0
⎤ ⎡ 0 0 ⎥ ⎢ 0⎦ + ⎣C2 I 0
⎡ I + σx 1 ⎢ = ⎣C2 + σy 4 0 ⎡
(1)
0 1/2 0 0 −1/2 0 0 0 0
(1)
C2 0 0
0 0 0 0 0 0 0 0 0
⎤ ⎡ 0 0 ⎥ ⎢ 0 ⎦ + ⎣σy σx 0
0 σx 0
⎤ 0 ⎥ 0 ⎦ I + σx + C2 + σy
C2 + σy I + σx 0 0 0 1/2 0 0 1/2 0 0 0
⎤ ⎡ σx 0 ⎥ ⎢ 0⎦ + ⎣0 C2 0
0 −1/2 0 0 1/2 0 0 0 0
0 0 1/2 0 0 1/2 0 0 0
0 0 0 0 0 0 0 0 0
⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎦ 1
0 0 0 0 0 0 0 0 0
P11 gives ⎡
0 ⎢ (1) u = ⎣0 0
0 −1 0
1 0 0
0 0 0
0 1 0
1 0 0
0 0 0
0 0 0
⎤ 0 ⎥ 0⎦ 1
σy 0 0
⎤ 0 ⎥ 0⎦ σy
148
Group action and symmetry projection operators
Similarly, we obtain (2)
1 [I + C2 4 ⎡ 2 0 ⎢0 0 ⎢ ⎢ 0 ⎢0 ⎢ ⎢−2 0 1 ⎢ = ⎢ 0 0 4 ⎢ ⎢0 0 ⎢ ⎢ 0 ⎢0 ⎢ ⎣0 0 0 0
P11 =
with (2)
# u = −1
0
− σx − σy ] 0 0 0 0 0 0 0 0 0
0
⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎦ 0
−2 0 0 2 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0
0
0
0
$ 0
0 0 −2 0 0 2 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 4 0
0 0 0
0 0 1
⎤ 0 ⎥ 0⎦ 0
1
and (3)
1 [I − C2 4 ⎡ 0 0 ⎢0 2 ⎢ ⎢ ⎢0 0 ⎢ ⎢0 0 1 ⎢ = ⎢ 0 2 4 ⎢ ⎢0 0 ⎢ ⎢ ⎢0 0 ⎢ ⎣0 0 0 0
P11 =
with
⎡
0 ⎢ (3) u = ⎣0 0 (4)
1 −1 0
+ σx − σy ] 0 0 2 0 0 −2 0 0 0
0 0 0
0 0 0 0 0 0 0 0 0
0 0 0
0 2 0 0 2 0 0 0 0
1 1 0
0 0 0
1 [I − C2 − σx + σy ] 4 ⎡ 2 0 0 2 0 0 0 ⎢0 0 0 0 0 0 0 ⎢ ⎢ ⎢0 0 0 0 0 0 0 ⎢ ⎢2 0 0 2 0 0 0 1 ⎢ = ⎢ 0 0 0 0 0 0 0 4 ⎢ ⎢0 0 0 0 0 0 0 ⎢ ⎢ ⎢0 0 0 0 0 0 4 ⎢ ⎣0 0 0 0 0 0 0 0 0 0 0 0 0 0
P11 =
0 0 0 0 0 0 0 0 0
⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎦ 0
⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎦ 0
6.2 Symmetry projection operators
149
with (4)
1 u= 0
0 0
0 0
1 0
0 0
0 0
0 1
0 0
0 0
From the rows of these projection matrices we obtain the following linearly independent symmetrized vectors: ⎧ ⎪ ⎪ |1, 1 = ⎪ ⎪ ⎨ (1) Γ : |1, 2 = ⎪ ⎪ ⎪ ⎪ ⎩|1, 3 = (2)
1 √ [0, 1, 0; 0, −1, 0; 0, 0, 0] , 2 1 √ [0, 0, 1; 0, 0, 1; 0, 0, 0] , 2 [0, 0, 0; 0, 0, 0; 0, 0, 1] ,
1 Γ : |2, 1 = √ [1, 0, 0; −1, 0, 0; 0, 0, 0] , 2
⎧ ⎪ ⎪ |3, 1 = ⎪ ⎪ ⎨ (3) Γ : |3, 2 = ⎪ ⎪ ⎪ ⎪ ⎩|3, 3 = ⎧ ⎨|4, 1 = (4) Γ : ⎩ |4, 2 =
1 √ [0, 1, 0; 0, 1, 0; 0, 0, 0] , 2 1 √ [0, 0, 1; 0, 0, −1; 0, 0, 0] , 2 [0, 0, 0; 0, 0, 0; 0, 1, 0] , 1 √ [1, 0, 0; 1, 0, 0; 0, 0, 0] , 2 [0, 0, 0; 0, 0, 0; 1, 0, 0] .
Molecular motion must include three degrees of freedom associated with the center-of mass (COM), and another three describing the rotations about the COM. The COM translation transforms as a vector; it engenders the three-dimensional vector Rep (v) Γ. The reduction of (v) Γ in terms of the C3v Irreps is (v)
Γ =(1) Γ +
(3)
Γ+
(4)
Γ.
Moreover, it requires that the constituent atoms have identical displacement vectors. Thus, we can write 6 $ 1 #√ 1 (v) 2 |4, 1 + |4, 2 , Γ, 1 := √ [1, 0, 0; 1, 0, 0; 1, 0, 0] = √ 3 3 6 $ 1 #√ 1 (v) 2 |3, 1 + |3, 3 , Γ, 2 := √ [0, 1, 0; 0, 1, 0; 0, 1, 0] = √ 3 3 6 $ 1 #√ 1 (v) 2 |1, 2 + |1, 3 . Γ, 3 := √ [0, 0, 1; 0, 0, 1; 0, 0, 1] = √ 3 3
150
Group action and symmetry projection operators
Comment on representations of angular momenta The angular momentum L of a particle about a point is given by L = r × p, where r and p are its position and linear momentum vectors, respectively. Since both r and p change sign under spatial inversion, L does not change sign under such operation. It is an axial vector – it transforms like a vector under proper ¯ as a combination of a proper rotations. We can consider an improper rotation R ¯ = RI = IR; and we find that rotation and an inversion, R (v)
¯ = − (v) χ(R). χ(R)
Since the determinants of proper and improper rotations in three dimensions are +1 and −1, respectively, the characters of an axial vector Rep can be derived from those of a polar vector Rep with the relation (a)
¯ = det |R| ¯ (v) χ(R). ¯ χ(R)
The characters for the three-dimensional axial vector Rep, of which the rotational angular momentum of the water molecule is a basis, are then given by χ(E) = 3,
χ(C2 ) = −1,
χ(σx ) = χ(σy ) = −1.
A reduction of this three-dimensional Rep in terms of the one-dimensional Irreps of C2v gives (a)
Γ =(2) Γ ⊕
(3)
Γ ⊕
(4)
Γ.
The corresponding symmetrized vectors are 6 1 (a) Γ, 1 = √ [1, 0, 0; −1, 0, 0; 0, 0, 0] = |2, 1 , 2 √ √ 6 1 1 3 3 1 (a) ; 0, , ; 0, −1, 0 0, , − Γ, 2 = √ 2 2 2 2 3 & 1 1 3 √ |3, 1 − = √ |3, 2 − |3, 3 , 2 3 2 6 1 (a) Γ, 3 = √ [−1, 0, 0; −1, 0, 0; 1, 0, 0] 3 $ √ 1 # = √ |4, 2 − 2 |4, 1 . 3 We can anticipate that, in the most general case, the matrix Γ(R) will be of dimension 3N × 3N , where N is the number of particles in the system. For the case of molecular vibration of the H2 O molecule, we found that it would be 9 × 9. A complete analysis of this system will be given in Chapter 15.
6.2 Symmetry projection operators
151
• The Wigner projection operator (µ) Pˆii can be used to project-out of a general function a basis function that transforms as the ith row of Irrep (µ). • The symmetry transfer operator (µ) Pˆij can then be used to generate all the partner basis functions. Program for generating projection operators for molecular vibrations The input information required is: (i) Generators of the point-group: • Irrep matrices GM, rotation matrices Rot, and permutations P of the point-group generators; • Number of generators NG, number of Irreps NC, and number of permutation entries. This is contained in the point-group file, for example C2v.m. (ii) Atomic position vectors are contained in RR[[i,j]], where i denotes either atomic species or orbit, and j, member atoms of species or orbit i. Thus, there are two ways to enter the required atomic positions data: you may record all atomic positions, classifying them by species, or you may enter orbit representatives, identifying them as first members of their respective orbits. In the latter case, you have to provide for the generation of the remaining members of each orbit with the aid of rotation matrices. (iii) The total number of atoms is NAT, the number of atomic species is NSP, or the number of orbits NOR. (iv) The number of atoms of species (or orbit) i is ATYPE[[i]]. (v) In case species are specified and not orbits, then orbits have to be generated. Ultimately, we classify the atomic positions into orbits. Sometimes different orbits may belong to the same elemental species. In the case of the H2 O molecule, the two H atoms occupy equivalent positions and fall in the same orbit, since they can be interchanged by the action of the operators of C2v .
Program MOLSYM Required input data and Mathematica packages: << C:/path/point-group.m << C:/path/mol.m << DiscreteMath‘Combinatorica’ << LinearAlgebra‘MatrixManipulation’
Group action and symmetry projection operators
152
1. Generate all the symmetry aspects of the point-group: Rotation matrices, Irrep matrices, and characters. Rot[1] = IdentityMatrix[3]; Initiate matrix Irreps in list Irrep, Irrep dimensions in dmu REP = {}; Do[ Switch[dmu[[i]] == 1, True, Do[AppendTo[REP, 1], {j, 1, g}], False, Do[ AppendTo[REP, IdentityMatrix[dmu[[i]]]], {j, 1, g} ] ], {i, 1, NC} ]; Irrep = Partition[REP, g]; L = {Range[NP]}; Do[ AppendTo[L, P[[i]]]; Do[ Irrep[[j, i + 1]] = GM[[j, i]], {j, 2, NC} ], {i, 1, NG} ]; Array[L, {NG + 1}]; nel = Length[L]; f := Permute[L[[i]], L[[j]]]; While[TrueQ[Length[L] < g], For[i = 1, i < g, i++, For[j = 1, j < (Length[L] + 1), j++, Switch[FreeQ[L, f], True, AppendTo[L, f]; nel++; Rot[nel] = Rot[i].Rot[j]; Do[ Switch[dmu[[mu]] == 1, True,
6.2 Symmetry projection operators
153
Irrep[[mu, nel]] = Irrep[[mu, i]] * Irrep[[mu, j]], ch[mu, nel] = Irrep[[mu, nel]], False, Irrep[[mu, nel]] = Irrep[[mu, i]] * Irrep[[mu, j]]; ch[mu, nel] = Tr[Irrep[[mu, nel]]] ], {mu, 2, NC} ] ] ] ] ] List Irrep now contains all matrix Irreps, for all group elements. List ch contains Irrep characters. 2. Symmetry of atomic positions: Operate on the atomic positions by group rotations and construct the permutations among equivalent atoms under the action of these operations. The permutation matrices are stored in PERMA. These matrices also define the molecule’s orbits. • Generate permutation matrices: PERMA = {Range[NAT]}; NPER = {}; Do[ NOA = 0; Do[ Do[ RP = Rot[kk] · RR[ik, jk]; kx = 0; park = 0; While[And[kx < ATYPE[[ik]], park == 0], kx++; RDIF = RP - RX[ik, kx]; Switch[RDIF . RDIF <0.01, True, lx = NOA + kx; AppendTo[NPER, lx]; park = 1 ] ], {jk, 1, ATYPE[[ik]]} ];
Group action and symmetry projection operators
154
NOA += ATYPE[[ik]], {ik, 1, NSM} ]; AppendTo[PERMA, NPER]; NPER = {}, {kk, 2, g}]; Pro is initially a 3NAT x 3NAT zero matrix Do[ Pro = {};
3. Construction of Irrep projection operators: ZRO = 0*IdentityMatrix[3]; Sympro = {}; Do[ dmux[imu] = 0, {imu, 1, NC} ]; Do[ AppendTo[Pro, ZRO], {i, 1, NAT}, {j, 1, NAT} ]; Pro = Partition[Pro, NAT]; • Construct projection operator P(mu, i) in Pro Do[ Do[ ic = PERMA[[kk, ir]]; Switch[dmu[[mu]] == 1, True, Pro[[ir, ic]]+ = ch[mu, kk] * Rot[kk], False, Pro[[ir, ic]]+ = Irrep[[mu, kk, nu, nu]] * Rot[kk] ], {kk, 1, g} ], {ir, 1, NAT} ]; Do[ Do[ Do[ im = 3* (ir - 1) + i;
6.2 Symmetry projection operators
155
Do[ jm = 3*(ic - 1) + j; Proj[im, jm] = dmu[[mu]] * Pro[[ir, ic, i, j]]/g, {j, 1, 3} ], {ic, 1, NAT} ], {i, 1, 3} ], {ir, 1, NAT} ]; Project = Array[Proj, {3*NAT, 3*NAT}]; EGV = Eigenvalues[Project]; EV = Eigenvectors[Project]; Do[ Switch[EGV[[ig]] == 1, True, dmux[mu]++; AppendTo[Sympro, EV[[ig]]] ], {ig, 1, 3*NAT} ], ",", {nu, 1, dmu[[mu]]} ],{mu, 1, NC} ] ]; Print[ "Symmetry-adapted eigenvectors: ", MatrixForm[Sympro] ]
6.2.3 The Irrep projection operator Often only the characters of Irreps are known. For this case it is useful to define yet another projection operator, the Irrep projection operator (µ) Pˆ as (µ) (µ) ˆ P = Pˆii i
dµ = g i =
dµ g ˆ R
where use has been made of (6.14).
(µ)
ˆ ˆ Γ−1 ii (R) R
ˆ R
(µ)
ˆ R, ˆ χ(R)
(6.21)
Group action and symmetry projection operators
156
Operating on a function Ψ, the Irrep projection operator produces a vector function in the subspace of the Irrep (µ) Γ given by
(µ)
Pˆ Ψ =
(µ)
i
=
dµ Pˆii Ψ = g
(µ)
ˆ R ˆΨ χ(R)
ˆ R
(µ)
ci
(µ)
(6.22)
φi .
i
This is a sum of the basis functions that span the subspace of the Irrep. Again, a unitary transformation U that diagonalizes (µ) P according to (6.18) does exist, and gives the symmetrized eigenvectors (µ) u that form a basis set for the subspace (µ), namely, (µ)
Pi (µ) u =(µ) u.
(6.23)
Example 6.9 Let us consider the action of the Irrep projection operator (3)
2 ˆ3 + (−1) C ˆ3−1 ˆ + (0)ˆ σ2 + (0)ˆ σ3 + (−1) C (2)E σ1 + (0)ˆ Pˆ = 6
of C3v on the functions of Example 6.3. We find ⎤ ⎡ ⎤ ⎡ ⎤⎞ ⎛ ⎡ ⎤ x x x x ⎢y⎥ ⎢y⎥⎟ ⎜ ⎢y⎥ ⎢y⎥ ⎢ ⎥ ⎢ ⎥⎟ ⎜ ⎢ ⎥ ⎢ ⎥ 1 ⎜ ˆ ⎢ ⎥ ˆ ⎢ ⎥ ˆ2 ⎢ ⎥ ⎟ ⎥ (3) ˆ ⎢ P ⎢ z ⎥ = ⎜2E ⎢ z ⎥ − C3 ⎢ z ⎥ − C3 ⎢ z ⎥ ⎟ ⎢ ⎥ ⎢ ⎥⎟ ⎢ ⎥ 3 ⎜ ⎢ ⎥ ⎣xz ⎦ ⎣xz ⎦ ⎠ ⎝ ⎣xz ⎦ ⎣xz ⎦ 2 2 2 x x2 x x √ √ ⎧ ⎤ ⎡ ⎤⎫ ⎡ −1 ⎤ ⎡ ⎪ ⎪ x + 23 y − 12√x − 23 y ⎪ ⎪ 2√ ⎪ 2x ⎥ ⎢ − 3 ⎥⎪ ⎪ ⎢ ⎪ 3 1 1 ⎪ ⎪ x − y x − y + ⎥ ⎥ ⎢ ⎪ ⎥ ⎢ ⎪ ⎢ ⎪ 2 2 2 2 ⎥ ⎢ ⎥⎪ ⎬ ⎨ ⎢ 2y ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ z z = ⎥ − ⎢ ⎥ ⎢ 2z ⎥ − ⎢ √ √ ⎥ ⎢ −1 ⎥⎪ ⎥ ⎢ −1 ⎢ 3 ⎪ 3 3 ⎪ ⎪ ⎪ ⎣2xz ⎦ ⎢ 2 x − 2 y z ⎥ ⎢ 2 x + 2 y z ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎦ ⎣ ⎣ √ √ ⎪ 2 2⎦⎪ ⎪ ⎪ 3 3 −1 −1 ⎭ ⎩ 2x2 x− y x+ y ⎡
2
⎡
2
⎤
2
2
⎤
⎡
x 3x ⎢ ⎥ ⎢ y 3y ⎢ ⎥ ⎢ 1 ⎢ ⎥ ⎢ 0 = 0 ⎢ ⎥=⎢ ⎥ ⎢ 3 ⎢ xz ⎣ ⎦ ⎣ 3xz √ 1 3 (x2 − y 2 ) + (x2 − y 2 ) + 3xy 2 2
⎥ ⎥ ⎥ ⎥. ⎥ ⎦ 1 √ xy 3
6.3 The regular projection matrices: the simple characteristic
157
⎤ ⎡ ⎤ ⎡ ⎤⎞ ⎛ ⎡ ⎤ x2 x2 x2 x2 1 ⎜ ˆ ⎢ 2 ⎥ ˆ ⎢ 2 ⎥ ˆ2 ⎢ 2 ⎥ ⎟ (3) ˆ ⎢ 2 ⎥ P ⎣y ⎦ = ⎝2E ⎣ y ⎦ − C3 ⎣ y ⎦ − C3 ⎣ y ⎦ ⎠ 3 xy xy xy xy ⎡
⎧ 2 ⎡ ⎤ √ ⎡ ⎤ −1 ⎪ x − 23 y 2 ⎪ 2 ⎪ 2x ⎪ ⎢ ⎥ ⎨⎢ 2 √ ⎥ ⎢ ⎥ 1 3 1 ⎢ 2y 2 ⎥ − ⎢ ⎥ = x − y ⎥ ⎣ ⎦ ⎢ 2 2 3 ⎪ ⎪ ⎣ √ ⎦ ⎪ √ ⎪ ⎩ 2xy −1 3 3 1 x − 2y x− 2 y 2 2 ⎢ ⎢ −⎢ ⎢ ⎣
−1 x 2
⎡ =
1 3
2 ⎤⎫ √ −1 ⎪ x + 23 y ⎪ 2 ⎥⎪ ⎪ 2 √ ⎥⎬ − 3 1 ⎥ x − 2y ⎥⎪ 2 √ ⎦ ⎪ ⎪ √ ⎪ ⎭ − 3 + 23 y x − 21 y 2
⎡
3 (x2 ⎢ 23 2 ⎢ (y ⎣2
− y2 ) + − x2 ) +
√ √
3xy
⎤
⎡
(x2 − y 2 ) +
7
1 xy 3
⎤
⎥ ⎥ ⎢ 7 ⎥ ⎢ 3xy ⎥ ⎦ = ⎣(y 2 − x2 ) + 1 xy ⎦. 3
1 xy 2
1 xy 3
We notice that in both cases, if we start with a pure basis function of (3) Γ, we end up with a pure basis function, and when the initial function is not properly symmetrized, such as x2 , we end up with a linear combination of the symmetry adapted basis functions of (3) Γ.
The Irrep projection operator obeys the idempotency property (µ)
Pˆ ∗
(ν)
Pˆ = δµν
(µ)
Pˆ ,
which is to be expected from a projection operator. In the case of one-dimensional Irreps, dµ = 1, the projection operator is called a primitive operator, because it projects onto a one-dimensional subspace. Since, in general, (µ) Pˆ will produce a general vector (function) in the irreducible space (µ), further resolution into row components has to be achieved by corresponding Wigner projection operators.
6.3 The regular projection matrices: the simple characteristic Next we construct the Wigner projection matrix for the Hilbert space of the regular Rep.
(µ)
dµ (reg) = Pˆii g
(µ)
ˆ Γ−1 ii (R)
(reg)
ˆ Γ(R),
(6.24)
ˆ R
which yields the (µ, i)th symmetry type. Note that here we have specified the general ˆ of (6.7) to be of a particular form, namely, the g × g regular Rep matrix. If we operator R
158
Group action and symmetry projection operators
sum over i we obtain a g ×g matrix projection operator called the simple characteristic, defined as (µ)
Pˆ (reg) =
(µ)
Pˆiireg .
(6.25)
i=1
This can be expressed in a more useful form as follows: (µ)
dµ Pˆ (reg) = g i =
dµ g
(µ)
ˆ Γ−1 ii (R)
(reg)
ˆ Γ(R)
ˆ R
(µ)
ˆ χ∗ (R)
(reg)
ˆ Γ(R)
ˆ R
⎧ ⎨ dµ (µ) ∗ χ (Ci ) = g C ⎩ i
=
(µ)
dµ g
Nc
(µ)
χ∗ (Ci )
(reg)
ˆ ∈ Ci R
(reg)
⎫ ⎬ ˆ Γ(R) ⎭
Ci .
i
Nc dµ Pˆ reg = g i
(µ)
χ∗ (Ci )
(reg)
Ci ,
(6.26)
where (reg) Ci is the regular Rep of the class matrix operator. This differs from (6.24) in ˆ has been replaced by the class matrix operator of the class to that the general operator R ˆ which R belongs and the summation is now over classes. It is clear from (6.26) that (µ) Pˆ reg is a g × g matrix that is a particular linear combination of the regular class matrices. Apart from an overall normalizing factor of dµ /g, the coefficient for each regular class matrix is the complex character of that class in the µth Irrep. Now, the elements of the regular class matrices are zeroes or ones as defined in (4.18) of Section 4.3, namely, ˆi , ˆj R ˆ −1 = R 1, if R (reg) k ˆ Γjk Ri = 0, otherwise. Thus, if we multiply a class matrix by its complex character, the new matrix consists of matrix elements that are either zero or the complex character of the class in the µth Irrep. Thus, from the definition of (µ) Pˆ reg in (6.26), we obtain the following simple expression for its elements: dµ (µ) ˆ −1 ˆ dµ (µ) (reg) (µ) ˆ reg χ Rl Rm = χ Γlm . (6.27) Plm = g g
6.3 The regular projection matrices: the simple characteristic
159
This is particularly useful for computational purposes. The simple characteristics of the different Irreps can be easily constructed from the irreducible characters determined by Dixon’s method. An example calculation also elucidates (6.26) and (6.27).
Example 6.10 Class matrices for C3v ⎡ 1 ⎢0 ⎢ ⎢0 ⎢ (reg) (reg) C1 = Γ(E) = ⎢ ⎢0 ⎢ ⎣0 0 (reg)
C2 =
(reg)
Γ(σ1 ) +
⎡
0 ⎢1 ⎢ ⎢0 ⎢ =⎢ ⎢0 ⎢ ⎣0 0 ⎡
1 0 0 0 0 0
0 ⎢0 ⎢ ⎢0 ⎢ + ⎢ ⎢1 ⎢ ⎣0 0 ⎡
0 ⎢1 ⎢ ⎢1 ⎢ =⎢ ⎢1 ⎢ ⎣0 0 (reg)
C3 =
(reg)
⎡ 0 ⎢0 ⎢ ⎢0 ⎢ = ⎢ ⎢0 ⎢ ⎣0 1
0 0 0 0 0 1 0 0 0 0 0 1
1 0 0 0 1 1
0 0 0 0 1 0
1 0 0 0 1 1
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
⎤ ⎡ 0 0 ⎢ 0⎥ ⎥ ⎢0 ⎥ ⎢ 1⎥ ⎢1 ⎥+⎢ 0⎥ ⎢0 ⎥ ⎢ 0⎦ ⎣0 0 0 ⎤ 0 0 0 1⎥ ⎥ ⎥ 1 0⎥ ⎥ 0 0⎥ ⎥ 0 0⎦ 0 0
0 0 0 1 0 0
⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 1
0 0 0 0 1 0
0 0 0 0 1 0
1 0 0 0 0 0
0 0 0 0 0 1
0 1 0 0 0 0
⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 1⎥ ⎥ 0⎦ 0
0 0 1 0 0 0
0 0 0 1 0 0
0 1 0 0 0 0
0 0 0 0 0 1
⎤ 1 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 0
⎤ 0 1⎥ ⎥ ⎥ 1⎥ ⎥ 1⎥ ⎥ 0⎦ 0
0 1 1 1 0 0
(reg)
0 0 1 0 0 0
Γ(σ2 ) +(reg) Γ(σ3 )
1 0 0 0 0 0
Γ(C3 ) + 0 0 0 1 0 0
(reg)
0 0 0 0 1 0 1 0 0 0 1 1
0 1 0 0 0 0
Γ(C32 )
1 0 0 0 0 0
⎤ ⎡ 0 0 ⎢ 0⎥ ⎥ ⎢0 ⎥ ⎢ 0⎥ ⎢0 ⎥+⎢ 0⎥ ⎢0 ⎥ ⎢ 1⎦ ⎣1 0 0
Group action and symmetry projection operators
160
⎡
0 ⎢0 ⎢ ⎢0 ⎢ = ⎢ ⎢0 ⎢ ⎣1 1
0 0 1 1 0 0
0 1 0 1 0 0
0 1 1 0 0 0
1 0 0 0 0 1
⎤ 1 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 1⎦ 0
The classes C2 and C3 are interchanged from Example 4.4, but note that it makes no difference how we label the classes, as long as we associate the correct character with each class. (µ) Pˆ reg is a sum of matrices and is independent of the labeling. For purposes of illustration we consider Irrep (3) Γ of the group C3v . The characters of the classes of are: (3)
(3)
χ(C1 ) = 2,
(3)
χ(C2 ) = 0,
χ(C3 ) = −1.
Using these and the regular class matrices we have just determined, we can use (6.26) to find the simple characteristic for Irrep (3): (3)
2 8 (2 × Pˆ reg = 6 ⎧⎡ 2 ⎪ ⎪ ⎪ ⎪ ⎢0 ⎪ ⎪ ⎢ ⎪ ⎨ ⎢0 1 ⎢ = ⎢ ⎢0 3⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎣0 ⎪ ⎪ ⎩ 0 ⎡
2 ⎢0 ⎢ 1⎢ ⎢0 = ⎢ 3⎢ 0 ⎢ ⎣−1 −1
(reg)
C1 ) + (0 ×
0 2 0 0 0 0
0 0 2 0 0 0
0 2 −1 −1 0 0
0 0 0 2 0 0
0 −1 2 −1 0 0
0 0 0 0 2 0 0 −1 −1 2 0 0
(reg)
C2 ) + (−1 ×
⎤ ⎡ 0 0 ⎢ 0⎥ ⎥ ⎢0 ⎥ ⎢ 0⎥ ⎢0 ⎥−⎢ 0⎥ ⎢0 ⎥ ⎢ 0⎦ ⎣1 1 2 −1 0 0 0 2 −1
0 0 1 1 0 0
0 1 0 1 0 0
0 1 1 0 0 0
(reg)
1 0 0 0 0 1
9 C3 ) ⎤ 1 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 1⎦ 0
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
⎤ −1 0 ⎥ ⎥ ⎥ 0⎥ ⎥ 0 ⎥ ⎥ −1⎦ 2
One sees from this example that, apart from a factor dµ /g, the matrix elements of the characteristic are just the characters of the corresponding elements of the regular multiplication table. This is expressed mathematically by (6.27), which is easily programmable and forms the basis for a computational determination of the simple characteristics. Following the procedure just illustrated, or using (6.27), which is much simpler, we find that ⎞ ⎛ 1 1 1 1 1 1 ⎜1 1 1 1 1 1⎟ ⎟ ⎜ ⎟ ⎜ 1 ⎜1 1 1 1 1 1⎟ (1) ˆ reg = P ⎟ ⎜ 6 ⎜1 1 1 1 1 1⎟ ⎟ ⎜ ⎝1 1 1 1 1 1⎠ 1 1 1 1 1 1
6.3 The regular projection matrices: the simple characteristic ⎛
(2)
Pˆ reg
⎜ ⎜ ⎜ 1 ⎜ = ⎜ 6 ⎜ ⎜ ⎝
1 −1 −1 −1 1 1
−1 1 1 1 −1 −1
−1 1 1 1 −1 −1
−1 1 1 1 −1 −1
1 −1 −1 −1 1 1
1 −1 −1 −1 1 1
161
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
The simple characteristics can be easily generated in Mathematica by the following function: Using the array Preg[mu,i,j] for the elements of the projection operators: we generate these elements as follows:
(µ)
Pˆ reg ,
Do[Preg[mu,i,j]=dm[mu]*charac[indc[m[[LI[[i]],j]]]]/g, {j,1,g}, {i,1,g}, {mu,1,nc}] where indc[k] is the class to which element k belongs, and LI[[i]] is the inverse of element i. Keeping in mind that Nc 1 nc(i) (µ) χ∗ (i) g i=1
(ν)
χ(i) = δµν ,
and (reg)
χ(i) =
Nc
dµ
(µ)
χ(i),
µ=1
where i stands for class i, we can now take the trace of the simple characteristic ˆ reg = Tr(µ) P
Nc dµ g i=1
(µ)
χ∗ (i) Tr Creg (i),
(6.28)
but the trace of the regular class-constant matrix is just the trace of the regular Rep of any element of the class multiplied by the number of elements in the class, nc(i). Thus, ˆ reg = Tr(µ) P
Nc dµ g i=1
=
Nc dµ g i=1
(µ)
χ∗ (i) nc(i) χreg (i)
(µ)
χ∗ (i) nc(i)
Nc
dν
(ν)
χ(i),
(6.29)
ν=1
where the last summation follows from recognizing that the trace of an element in the regular Rep is the sum of the characters for each Irrep multiplied by the multiplicity of that Irrep in the regular Rep. Thus (6.29) can be rewritten as ˆ reg = Tr(µ) P
Nc Nc dµ dν g ν=1 i=1
(µ)
χ∗ (i) nc(i)
(ν)
χ(i).
(6.30)
162
Group action and symmetry projection operators
Using the orthogonality theorem for characters, (4.34), we now have ˆ reg = d2µ . Tr(µ) P
(6.31)
The simple characteristics are, therefore, singular matrices of rank d2µ . As projection operators they are also idempotent and Hermitian, ((µ) Pˆ reg )2 =(µ) Pˆ reg , and the columns are eigenvectors with unit eigenvalues. This provides a check that the calculated simple characteristic is probably correct.
Example 6.11 Direct matrix multiplication demonstrates that (3) Pˆ reg from Example 6.10 is idempotent. Operating on the first column of (3) Pˆ reg by (3) Pˆ reg itself, we find that the column is an eigenvector with unit eigenvalue: ⎤⎡ ⎤ ⎡ 2 2 0 0 0 −1 −1 ⎥⎢ 0 ⎥ ⎢ 0 2 −1 −1 0 0 ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ 2 −1 0 0 ⎥⎢ 0 ⎥ 1 ⎢ 0 −1 ⎥⎢ ⎥ ⎢ 3 ⎢ 0 −1 −1 2 0 0 ⎥⎢ 0 ⎥ ⎥⎢ ⎥ ⎢ ⎣−1 0 0 0 2 −1⎦ ⎣−1⎦ −1 −1 0 0 0 −1 2 ⎤ ⎡ ⎤ 2 6 ⎢ 0⎥ ⎢ 0⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ 0⎥ 1 ⎢ ⎢ ⎥ ⎢ 0⎥ = ⎢ ⎥ = 1⎢ ⎥ ⎢ 0⎥ 3 ⎢ 0⎥ ⎢ ⎥ ⎢ ⎥ ⎣−1⎦ ⎣−3⎦ −1 −3 ⎡
Since idempotency requires that the columns of (µ) Pˆ reg are also eigencolumns of (µ) Pˆ reg , with unit eigenvalues, it follows that the order of the set of linearly independent eigencolumns would be given by its trace, i.e. d2µ . Consequently, we should be able to find a rectangular reduction matrix (µ) Tˆ of order g × d2µ by looking for d2µ orthogonalized eigencolumns of (µ) Pˆ reg using a Schmidt orthogonalization procedure, which can be simplified by the use of the idempotency and hermiticity of (µ) Pˆ reg .
Exercises 6.1 Determine the orbits, stabilizers, and strata of the action of 1 0 1 0 1− 0 −1 G := , , , 0 1 0 −1 0 1 0 on the xy-plane.
* 0 , , −1
Exercises
163
6.2 With reference to Example 6.3, to which rows of (3) Γ do the functions {yz, xy} belong, if any? 6.3 The valence electron orbitals of a water molecule consist of one 1s-orbital on each H atom, and the three-fold degenerate 2p-orbital manifold centered on the O atom. Under the C2v symmetry group operations, the permutations among the atoms are the same as those considered in Example 6.8. However, the function-space is now different as it consists of electron wavefunctions. (i) Determine the Rep engendered by C2v on the set of electron states. (ii) Derive the symmetry-adapted states of the water molecule. 6.4 The ammonia molecule NH3 has C3v symmetry. Determine: (i) Its symmetry-adapted vibrational modes. (ii) Its symmetry-adapted molecular orbitals. (Again, consider s-orbitals centered on the H atoms, and a p-manifold on the N atom. In the following problems we consider molecules which contain carbon atoms. The four valence electrons of a carbon atom occupy both the 2s and 2p states, which have to be included in each set of orbitals of these molecules. 6.5 Repeat Exercise 6.4. for the case of a planar molecule of the form AB3 , such as CO3 2− , which has D3h symmetry. 6.6 In the methane molecule, CH4 , the C atom is located at the center of a tetrahedron, while the H atoms are at its apices. Repeat Exercise 6.4 for this molecule. 6.7 Repeat Exercise 6.4. for the benzene molecule. It consists of six carbon atoms forming the apices of a hexagon and six hydrogen atoms bound radially, thus having D6h symmetry.
7 Construction of the irreducible representations
In this chapter we use the method of projection operators, presented in Chapter 6, to develop a general procedure for calculating all the Irreps of an arbitrary finite group. The method was first proposed by Stig Flodmark and Esko Blokker [6, 7, 8, 9, 10, 11]. The essence of this method is to establish a general procedure for obtaining the matrices of an Irrep (α) from the matrices of the regular Rep with the aid of projection matrices. Thus, for an Irrep of dimension dα , we need to construct a matrix (α) Φ of dimension g ×dα , such that when it operates on a matrix (reg) Γ(R) of the regular Rep we obtain (α) Γ(R), namely, (α)
Φ† (reg) Γ(R) (α) Φ =
(α)
Γ(R),
∀R ∈ G.
(7.1)
The construction of such a projection matrix is achieved by using the simple characteristic matrix of the Irrep α, introduced in the preceeding chapter, to project dα linearly independent vectors from the eigenvectors of the regular Rep matrices. In the remaining part of this chapter, we describe how to utilize the permutative character of the regular Rep matrices to obtain their eigenvector, and how to project out the symmetrized eigenvector components that belong to the subspace of the Irrep α.
7.1 Eigenvectors of the regular Rep A close examination of the matrices of the regular Rep reveals that they have the simple structure of permutation matrices. This is no surprise, when we consider the structure of the regular matrices in terms of the rearrangement theorem. As a consequence, their eigenvectors can be simply produced through determining their cycle structure. ˆ k ), then the operation If uλ is an eigenvector of (reg) Γ(R (reg)
ˆ k ) uλ = u = λ u λ Γ(R
(7.2)
yields a vector u in which the elements, (u1 , u2 , . . . , ug ), are permuted, and such that ui = ui /λ.
(7.3)
ˆ k means that the permutations are partitioned into Moreover, the cycle structure of R subsets, where the elements of each subset are permuted only among themselves. We also should recall that group elements in the same class have the same cycle structure. 164
7.1 Eigenvectors of the regular Rep
165
ˆ k ) by choosing the only nonzero We can construct independent eigenvectors of (reg) Γ(R ˆk . elements of uλ to be the subset corresponding to one of the permutation cycles of R And, according to (7.2), when we choose the first element of such a subset to be equal to one, all others must equal powers of λ. The eigenvalue λ may assume any of the powers of ξ = e(2πi/cycle ) , where cycle is the permutation cycle length. Since n, the group exponent, is a divisor of cycle , ξ = ζ n/cycle . Each allowed value has a corresponding eigenvector in which only the elements corresponding to one permutation cycle are different from zero and equal to powers of λ. Working through all powers of ζ and all cycles, one can quickly construct a complete set of g linearly independent eigencolumns. That is already more than we need. We need choose only one eigenvalue λ and construct its eigencolumns.
Example 7.1 Construction of eigencolumns belonging to
(reg)
ˆ 1 ) of C 3v Γ(σ
From Example 4.1 we know that ⎛
0 ⎜1 ⎜ ⎜0 ⎜ (reg) Γ(ˆ σ1 ) = ⎜ ⎜0 ⎜ ⎝0 0 Let this matrix Rep operate ⎛ 0 ⎜1 ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎜ ⎝0 0
1 0 0 0 0 0
0 0 0 0 0 1
0 0 0 0 1 0
0 0 0 1 0 0
⎞ 0 0⎟ ⎟ ⎟ 1⎟ ⎟. 0⎟ ⎟ 0⎠ 0
on a general column vector, yielding ⎞⎡ ⎤ ⎡ ⎤ u1 1 0 0 0 0 u2 ⎢ ⎥ ⎢u 1 ⎥ 0 0 0 0 0⎟ ⎟ ⎢u 2 ⎥ ⎢ ⎥ ⎟⎢ ⎢u ⎥ 0 0 0 0 1⎟ ⎢u3 ⎥ ⎥ ⎢ 6⎥ ⎟ ⎢ ⎥ = ⎢ ⎥. ⎢u 5 ⎥ 0 0 0 1 0⎟ ⎢u4 ⎥ ⎟⎢ ⎥ ⎢ ⎥ ⎣u 4 ⎦ 0 0 1 0 0⎠ ⎣u5 ⎦ 0 1 0 0 0 u6 u3
σ1 ) operating on a vector is to permute the elements We see that the effect of (reg) Γ(ˆ among themselves according to the permutation cycles (12)(36)(45). The element σ1 is of order 2, so are its cycle lengths, so the possible eigenvalues are powers of ξ = e(2πi/2) , namely, λ = ±1. It is clear then that there exist three independent eigencolumns, for each eigenvalue, of the form ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 1 u1 0 0 ⎢u2 ⎥ ⎢λ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢u ⎥ ⎢ 1 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 3⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥, ⎢ ⎥ = ⎢ ⎥, ⎢ ⎥ = ⎢ ⎥, ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢u 4 ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣u5 ⎦ ⎣λ⎦ 0 0 λ 0 u6 0
166
Construction of the irreducible representations
if we choose the first element to be unity. Letting λ take both its values, ±1, we obtain the following six eigenvectors, which span the six-dimensional space of C3v : ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 1 0 0 1 ⎢1⎥ ⎢0⎥ ⎢0⎥ ⎢−1⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢0⎥ ⎢1⎥ ⎢0⎥ ⎢ 0 ⎥ ⎢ 1 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥. ⎢0⎥ ⎢0⎥ ⎢1⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣0⎦ ⎣0⎦ ⎣1⎦ ⎣ 0 ⎦ ⎣ 0 ⎦ ⎣−1⎦ 0 −1 0 0 1 0 Let’s try one more. What happens if we choose ⎛ 0 0 ⎜0 0 ⎜ ⎜0 0 ⎜ (reg) ˆ Γ(C3 ) = ⎜ ⎜0 1 ⎜ ⎝0 0 1 0 Letting this operate on a general ⎛ 0 0 ⎜0 0 ⎜ ⎜0 0 ⎜ ⎜ ⎜0 1 ⎜ ⎝0 0 1 0
C3 ? We know from Example 4.1 that ⎞ 0 0 1 0 1 0 0 0⎟ ⎟ ⎟ 0 1 0 0⎟ ⎟. 0 0 0 0⎟ ⎟ 0 0 0 1⎠ 0 0 0 0
vector we find ⎞⎡ ⎤ ⎡ ⎤ u1 0 0 1 0 u5 ⎢ ⎥ ⎢ u3 ⎥ 1 0 0 0⎟ ⎟ ⎢u2 ⎥ ⎢ ⎥ ⎟⎢ ⎢u ⎥ 0 1 0 0⎟ ⎢u3 ⎥ ⎥ ⎢ 4⎥ = ⎟⎢ ⎥ ⎢ ⎥, ⎢ u2 ⎥ 0 0 0 0⎟ ⎢u4 ⎥ ⎟⎢ ⎥ ⎢ ⎥ ⎣ u6 ⎦ 0 0 0 1⎠ ⎣u5 ⎦ 0 0 0 0 u6 u1
corresponding to the same permutation cycle structure (156)(234) that we found in Example 2.1. This gives us two independent eigencolumns. Since C3 is of order 3, possible eigenvalues are powers of ζ = e(2πi/3) . Two independent eigencolumns, with three different eigenvalues for each, once again yields six eigencolumns to span the six-dimensional space of C3v . They are: ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ ⎡ 0 1 1 0 0 1 ⎢ 0 ⎥ ⎢ 1 ⎥ ⎢ 0 ⎥ ⎢ 1 ⎥ ⎢0⎥ ⎢1⎥ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎢ 0 ⎥ ⎢ei(2π/3) ⎥ ⎢ 0 ⎥ ⎢ei(4π/3) ⎥ ⎢0⎥ ⎢1⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥. ⎥, ⎢ ⎥, ⎢ ⎢ ⎢ 0 ⎥ ⎢ei(4π/3) ⎥ ⎢ 0 ⎥ ⎢ei(2π/3) ⎥ ⎢0⎥ ⎢1⎥ ⎥ ⎢ i(4π/3) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ i(2π/3) ⎥ ⎢ ⎦ ⎣ 0 ⎦ ⎣e ⎦ ⎣ 0 ⎦ ⎣1⎦ ⎣0⎦ ⎣e 0 1 0 0 ei(4π/3) ei(2π/3)
7.1.1 Computer generation of eigenvectors of the regular Rep ˆ k ) is based on determining its The algorithm used in generating an eigenvector of (reg) Γ(R cycle structure when considered as the permutation of g objects. Each cycle generates a number of eigenvectors equal to the cycle length, with corresponding eigenvalues λ equal to powers of the primitive root ξ = e(2πi/cycle ) . We select only one of these eigenvectors. It is constructed by setting nonzero values, only at the locations of the cycle elements: placing
7.1 Eigenvectors of the regular Rep
167
1 in the first location of the given cycle, and assigning to the remaining locations of the cycle powers of the eigenvalue λ. Notice that the values of λ are confined to the set of powers of ζ = e(2πi/n) , where n is the group exponent. (i) To determine the cycle structure, we make use of the fact that the nonzero locations, ˆ k ) satisfy the relation R ˆ −1 R ˆ j . Hence, in effect, when (reg) Γ(R ˆk ) ˆi = R ij, in (reg) Γ(R k operates on a vector it replaces the entry in the i location by that in the j location. ˆ k as the Since the elements of a class have the same cycle structure, we choose R first element in each class. The following program generates the eigenvectors for the representative element LC[[i,1]] of class i. Cycle lengths are stored in Lcycle, the cycle structures in loops, and number of cycles in nocycl[i-1]. lpl={}; Lcycle={} inv=LI1[[LC[[i,1]]]];plp=0; Do[flag[j]=0,{j,1,g}]; first=1;llp=0;cnt=1;elm=0; While[ And[first<=g,cnt<=g], While[And[first
1,True,iv=iv+Lcycle[[i-1,nov-1]]]; Do[rvec[nov,i]=0,{i,1,g}]; rvec[nov,iv]=1;nrt=1;pp=iv+Lcycle[[i-1,nov]];
Construction of the irreducible representations
168
j=iv;ex=Exp[ı(2π/Lcycle[[i-1,nov]])]; While[j
7.2 The symmetry structure of the regular Rep eigenvectors Suppose we find a unitary transformation that produces an Irrep equivalent to the Irrep ˆ k ) of the specific group operator R ˆ k is Γ where only the matrix representative (α) Γ(R found in diagonal form, i.e. with eigenvalues, (α) λ1 , (α) λ2 , . . . , (α) λdα . (Note that no claim ˆ i ), i = k, are diagonal.) Then, recalling from our approach is made that any other (α) Γ(R ˆ k ∈ G, to Dixon’s method that if n is the exponent of the group G, with element R (α)
ˆ k )n = E. ˆ (R Thus, we must have
(α)
n
ˆk ) Γ(R
from which we conclude that each eigenvalue (ζ 0 , ζ 1 , ζ 2 , . . . , ζ n−1 ) with
= (α)
(α)
ˆ Γ(E),
ˆ k ) will be one of the numbers λk of (α) Γ(R
ζ = e(2πi/n) . Since the regular Rep is completely decomposable into group Irreps, it is also true that ˆ k ) is equal to one of the numbers (ζ 0 , ζ 1 , ζ 2 , . . . , ζ n−1 ). each of the eigenvalues λ of (reg) Γ(R We seek a transformation of the regular Rep that reduces it completely, not just to block-diagonal form, but to diagonal form. Of course, each Irrep (α) Γ occurs dα times along the main diagonal, but we even assume that the dα matrices of each Irrep (α) along the main diagonal are exactly equal, not just equivalent. We denote this special g × g Rep ˆ k ) to distinguish it from the ordinary regular Rep (reg) Γ. by (reg) ΓD (R
Example 7.2 ˆ 3 ) for C 3v The matrix (reg) ΓD (C It would be of the form
⎛
⎜ ⎜ ⎜ ⎜ (reg) ˆ ΓD (C3 ) = ⎜ ⎜ ⎜ ⎝
1 0 0 0 0 0
0 1 0 0 0 0
0 0 i(2π/3)
e
0 0 0
0 0 0 e−i(2π/3) 0 0
0 0 0 0 ei(2π/3) 0
0 0 0 0 0 e−i(2π/3)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
7.2 The symmetry structure of the regular Rep eigenvectors
169
The space (reg) V of the g × g Rep (reg) ΓD is spanned by g vectors. The space comprises dα identical irreducible subspaces associated with each Irrep (α) of the group G. We write the basis vectors of the ith subspace of Irrep (α), (α) V i , as (α) V i such that ⎡ (α) i ⎤ v1 ⎢ (α) v i ⎥ ⎢ 2 ⎥ ⎥ ˆk ⎢ ˆ k (α) V i = R R ⎢ ... ⎥ ⎢ ⎥ ⎣ ... ⎦ (α) i vd α = [(α) v1i ,
(α) I v2 ,
...,
(α) i ˆ k ), vdα ] (α) Γi (R
(7.4)
for i = 1, 2, . . . , dα . To reiterate, we note that the Irrep (α) Γ is of dimension dα . Since each Irrep of dimension dα appears dα times along the diagonal of the regular Rep, there are dα identical subspaces. Thus, (7.4) holds for i = 1, 2, . . . , dα . For the complete g-dimensional space of (reg) ΓD we have ˆ k [ (1) V 1 , R
(1)
V 2, . . . ,
= [ (1) V 1 , . . . ,
(nc)
where the sets of basis vectors number of classes.
(1)
V d1 ,
V dnc ]
(α)
(2)
(reg)
V 1, . . . ,
(nc)
V 1, . . . ,
(nc)
V dnc ]
ˆ k ), ΓD (R
(7.5)
V i range over all Irreps, of which there are nc, the
Example 7.3 ˆ3 of the group C3v , which has three classes and hence three Irreps, of For the element C which two are one-dimensional and one is two-dimensional, (7.5) becomes ˆ3 [ (1) V 1 , C
(2)
V 1,
(3)
V 1,
(3)
V 2 ] = [ (1) V 1 ,
(2)
V 1,
(3)
V 1,
(3)
ˆ3 ). V 2 ] (reg) ΓD (C
ˆ k ), the matrix representative of R ˆ k in (7.5), is diagonal, not just We assume that (α) Γ(R (α) ˆ block diagonalized. We also assume that Γ11 (Rk ) is a nondegenerate eigenvalue λ, so ˆ k operates on the first basis vector the eigenvalue equation is simply when R ˆk R
(α) i v1
= λ
(α) i v1 ,
(7.6)
for i = 1, 2, . . . , dα . Because an Irrep of dimension dα occurs dα times in the regular Rep, ˆ k ) or (reg) Γ(R ˆ k ). It so must the eigenvalue λ occur at least dα times for either (reg) ΓD (R R ˆ (reg) ˆ k ), as other Irreps (β) Γ(R ˆ k ) can occurs probably even more times in Γ (Rk ) and Γ(R also have this eigenvalue, even several times, since for other Irreps we do not require that λ be nondegenerate. ˆ k ), belonging to eigenvalue λ, is then The general form of the eigenvector of (reg) Γ(R vλ =
dα i=1
ai
(α),i
v1 +
dβ β=α i=1
k
bβik
(β),i
vk ,
(7.7)
Construction of the irreducible representations
170
ˆk ) where the summation in the second term runs over all β, (β = α) for which (β) Γ(R has eigenvalue λ, and the summation over k is to allow for degenerate eigenvalues λ in (β) ˆ k ). We find that contributions from the last term in (7.7) disappear in the process Γ(R of acting on vλ with the Irrep projection operator (α) Pˆ . Thus there is no need for the reader uninterested in every mathematical fine point to be concerned with this term.
7.3 Symmetry projection on regular Rep eigenvectors A symmetrized vector (α) Vλ in the subspace of the Irrep (α), is now obtained by applying the Irrep projection operator, (α)
g ˆ = dα P g m=1
(α)
ˆ m) R ˆ m, χ∗ (R
(7.8)
to uλ , since all contributions of the second term in (7.7) are canceled, and we obtain (α)
ˆ uλ = P
dα
(α) i v1
ai
=
(α)
w1 .
(7.9)
i=1
The nondegeneracy of λ is easily established. If the number of independent vectors obtained after projection of the independent eigenvectors of the eigenvalue λ is larger than dα , then λ is degenerate and one must choose another eigenvalue or another group element to start with. We then find that ˆm R
(α)
ˆm w1 = R
dα
ai
(α) i v1
=
dα
(α)
ws
αi
(α) i vs ,
(α)
ˆ m )s1 = Γ(R
(α)
φm ,
(7.10)
s=1
i=1
with (α)
ws =
dα
(7.11)
i=1
ˆ m = R ˆ k . (Recall that R ˆ k is the specific element of the previous sections with where R (α) ˆ diagonal Rep ΓD (Rk ).) So the vectors (α) ws , s = 1, 2, . . . , dα also span an irreducible (α) space Ω, transforming according to (α) Γ. The vectors (α) φ1 ≡ (α) w1 , and (α) φm lie in ˆ m (m = 2, 3, . . . , g) on (α) φ1 = the space (α) Ω. By operating with different elements R (α) w1 , we finally obtain dα linearly independent vectors (α) φd , d = 1, 2, . . . , dα and these form a basis (α) Φ = ((α) φ1 , . . . ,(α) φdα ) of (α) Ω. The linear independence is checked by orthonormalizing each newly produced (α) φd to the previous ones by Schmidt’s procedure. Since this basis of (α) Ω is just as good as the basis (α) W = ((α) w1 , . . . ,(α) wdα ) it will transform as (α) Γ: ˆp R
(α)
Φ =
(α)
Φ
(α)
ˆ p ), Γ(R
(7.12)
ˆ p ) differ from those corresponding to (α) W only by a linear transforThe matrices (α) Γ(R mation. Since we constructed (α) Φ as an orthonormal basis, we immediately obtain the
7.3 Symmetry projection on regular Rep eigenvectors unitary Irrep
(α)
171
Γ: (α)
ˆp ) = Γ(R
(α)
Φ†
(reg)
Γ(Rp )
(α)
Φ,
(7.13)
where Φ is a g × dα rectangular matrix.
Example 7.4 Finding the two-dimensional Γ Irrep of C3v We begin by finding the orthonormal basis (3) Φ that belongs to the subspace of the two-dimensional Irrep (3) Γ of the group C3v . From Example 6.6 we know the simple characteristic of the two-dimensional Irrep of the group C3v is ⎞ ⎛ 2 0 0 0 −1 −1 ⎜0 2 −1 −1 0 0 ⎟ ⎟ ⎜ ⎟ ⎜0 −1 2 −1 0 0⎟ 1 ⎜ (3) ˆ reg = P ⎟. ⎜ 3 ⎜0 −1 −1 2 0 0 ⎟ ⎟ ⎜ ⎝−1 0 0 0 2 −1⎠ −1 0 0 0 −1 2 (3)
If we were to operate on each of the eigencolumns from Example 7.3 with (3) Pˆ reg , we would find that the first set of six eigencolumns do not belong to the subspace of the two-dimensional Irrep of (3) Γ. This simple exercise is left to the reader. However, if we choose the first eigencolumn from the second set of six eigencolumns we find ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 1 1 2 0 0 0 −1 −1 ⎜ ⎟ ⎜ 0 ⎟ ⎜ 0 2 −1 −1 0 0 ⎟ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ 0 −1 2 −1 0 0⎟ ⎜ 1 ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ = ⎜ ⎟, ⎜ ⎜ 0 ⎟ 3 ⎜ 0 −1 −1 2 0 0 ⎟ ⎜ 0 ⎟ ⎟ ⎜ i(2π/3) ⎟ ⎜ i(2π/3) ⎟ ⎜ ⎠ ⎝e ⎠ ⎝−1 0 0 0 2 −1⎠ ⎝e i(4π/3) i(4π/3) −1 0 0 0 −1 2 e e thus demonstrating that we have found an eigencolumn belonging to (3) Γ. However, there is some ambiguity in the last set of eigencolumns in Example 7.1, because columns 1 and 3, as well as columns 2 and 4, are not orthogonal to each other. This is because they belong to the doubly degenerate two-dimensional Irrep (3) Γ. There is a systematic procedure that we can adopt in order to find the second eigencolumn that we need to span the two-dimensional subspace of (3) Γ. We operate on the first eigencolumn with other operations of the group in order to generate a partner eigencolumn. We choose (reg) Γ(ˆ σ1 ), which we know from Example 7.2, obtaining ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 1 0 1 0 0 0 0 0 ⎜1 0 0 0 0 0⎟ ⎜ 0 ⎟ ⎜ 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜0 0 0 0 0 1⎟ ⎜ 0 ⎟ ⎜ei(4π/3) ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ = ⎜ i(2π/3) ⎟. ⎟ ⎜0 0 0 0 1 0⎟ ⎜ 0 ⎟ ⎜e ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎝0 0 0 1 0 0⎠ ⎝ei(2π/3) ⎠ ⎝ 0 ⎠ 0 0 1 0 0 0 0 ei(4π/3)
Construction of the irreducible representations
172
We now have an eigencolumn that is independent of and orthogonal to the first, and which is a partner to it. These two eigencolumns thus form a basis for the two-dimensional Irrep (3) Γ, which we can normalize and write as ⎛
1 0 0 0
⎜ ⎜ ⎜ 1 ⎜ (3) Φ = √ ⎜ 3 ⎜ ⎜ i(2π/3) ⎝e ei(4π/3)
⎞
0 1
⎟ ⎟ e ⎟ ⎟. ei(2π/3) ⎟ ⎟ 0 ⎠ 0 i(4π/3) ⎟
Now we can generate the full two-dimensional Irrep (3) Γ by following the prescription of (7.13) for each element of the group. We find for (3) Γ(ˆ σ1 ), with (3)
1 0
1 Φ† = √ 3
0 1
0
0
e−i(4π/3)
e−i(2π/3)
e−i(2π/3) 0
e−i(4π/3 , 0
that (3)
Γ(ˆ σ1 ) =
(3)
† (reg)
Φ
Γ(ˆ σ1 )
(3)
0 1
Φ =
1 . 0
Continuing in this fashion for all other elements of the group, we find: (3)
ˆ = Γ(E)
1 0
(3)
Γ(ˆ σ2 ) =
0 , 1 0
ei(2π/3)
(3)
e−i(2π/3) , 0
(3)
ˆ3 ) = Γ(C
ei(2π/3) 0
Γ(ˆ σ1 ) =
e−i(2π/3)
1 , 0
(3)
Γ(ˆ σ3 ) =
(3)
ˆ32 ) = Γ(C
0 e−i(2π/3)
ei(2π/3) , 0
0
0 1
,
e−i(2π/3) 0
0 ei(2π/3)
.
7.4 Computer construction of Irreps with dα >1 (i) Normalized symmetry-projected basis vector The simple characteristic matrix defined in Chapter 6 is used to construct the symmetry-projected vector svec, which belongs to the nrep Irrep, from the eigenvector rvec of the regular Rep. After normalization it is stored as the first column of the g×dnrep matrix symmp.
7.4 Computer construction of Irreps with dα >1
173
Do[svec[i]=0,{i,1,g}]; Do[ elm=m[[1,i,LI[[j]]]]; elm=indc[elm];svec[i]+=charac[nrep,elm]*rvec[j], {i,1,g},{j,1,g} ]; norm=Sum[svec[i]*Conjugate[svec[i]],{i,1,g}]; √ Do[symmp[i,1]=svec[i]/ norm,{i,1,g}]
(ii) Construction of remaining Irrep basis vectors by application of other group elements Since group elements that are powers of the element k will have the same eigenvector of the regular Rep, they should be ignored in this procedure. This is implemented by an array ban which assigns a value 1 to such group elements.
Do[ban[i]=0,{i,1,g}]; ban[1]=1;ban[LC[[class,1]]]=1;ix=1; elmpwr=LC[[class,1]];expo=LC1[[class]]-1; While[ix<expo, elmpwr=m[[1,elmpwr,LC[[class,1]]]]; ban[elmpwr]=1,ix++ ]; nsymm=1;dl=dim[nrep];iy=1; While[nsymm
174
Construction of the irreducible representations
Do[scalar+=Conjugate[symmp[j,i]]*svec[j], {j,1,g}]; Switch[Abs[scalar]<0.001, False, Switch[Abs[scalar-1]<0.001, True,iy++;Goto[Try], False, Do[ svec[kk]=svec[kk]-scalar*symmp[kk,i], {kk,1,g}]; norm=0; Do[ norm+=svec[kk]*Conjugate[svec[kk]], {kk,1,g}]; √ norm= norm; Switch[norm<0.001, True,Goto[Try] ]; Do[svec[kk]=svec[kk]/norm, {kk,1,g}] ] ], {i,1,nsymm}]; nsymm++; Do[symmp[j,nsymm]=svec[j],{j,1,g}] ];
(iii) Generate the Irrep using (13)
Do[hermsymp[i,j]=Conjugate[symmp[j,i],{i,1,dl},{j,1,g}]; temps={}; Do[rep[j,k]=0,{j,1,dl},{k,1,dl}];
7.4 Computer construction of Irreps with dα >1
Do[Switch[j==k,True,rep[j,k]=1],{i,1,dl},{j,1,dl}]; AppendTo[temps,Irrep=Array[rep,{dl,dl}]; Do[rep[j,k]=0,{j,1,dl},{k,1,dl}]; Do[ Do[ grp=m[[1,LI[[iv]],j]]; Do[ gensym[j,k]=symmp[grp,k],{k,1,dl} ],{j,1,g} ]; Do[ Do[ rep[i,j]+=hermsymp[i,k]*gensym[k,j],{k,1,g}], {i,1,dl},{j,1,dl} ]; Irrep=Array[rep,{dl,dl}]; AppendTo[temps,Irrep]; Do[rep[j,k]=0,{j,1,dl},{k,1,dl}], {iv,2,lgen} ]; Do[ban[i]=0,{i,1,g}]; Do[ban[i]=1,{i,1,lgen}];jx=2; While[Sum[ban[i],{i,g}]
175
Construction of the irreducible representations
176
Irrep=Array[rep,{dl,dl}]; AppendTo[temps,Irrep]; Do[rep[j,k]=0,{j,1,dl},{k,1,dl}]; ban[mm]=1]; jx++]; AppendTo[gamma,temps];temps={} ]
7.5 Summary of the method • Form a nondegenerate eigencolumn of
(reg)
• Operate on it with the projection matrix
ˆ k ). Γ(R reg Pˆ .
(α)
ˆ m ) and orthonormalize, • Operate on the projected column with different matrices Γreg (R (α) obtaining dα orthonormal columns, which form the matrix Φ. • Generate the Irrep (α) according to (7.13). We illustrate the method with another example.
Example 7.5 The group C4v The multiplication table is E 1 2 3 4 5 6 7 8
C4 2 3 4 1 7 8 6 5
C2 3 4 1 2 6 5 8 7
C43 4 1 2 3 8 7 5 6
σx 5 8 6 7 1 3 4 2
σy 6 7 5 8 3 1 2 4
σdxy 7 5 8 6 2 4 1 3
σdx¯y 8 6 7 5 4 2 3 1
It has a two-dimensional Irrep with characters (2, 0, −2, 0, 0, 0, 0, 0). According to (6.27) ⎛
(5)
Pˆ reg
1 ⎜ 0 ⎜ ⎜−1 ⎜ ⎜ 0 ⎜ = (1/2) ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 0 0
0 1 0 −1 0 0 0 0
−1 0 1 0 0 0 0 0
0 −1 0 1 0 0 0 0
0 0 0 0 1 −1 0 0
0 0 0 0 −1 1 0 0
0 0 0 0 0 0 1 −1
0 0 0 0 0 0 −1 1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠
7.5 Summary of the method
177
ˆ4 is of order 4, so ζ = exp(2πi/4) = i and possible eigenvalues are (1, i, −1, −i). Element C ⎛ 0 ⎜1 ⎜ ⎜0 ⎜ ⎜0 (reg) ˆ4 ) = ⎜ Γ(C ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
⎞ 0 0⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟. 0⎟ ⎟ 1⎟ ⎟ 0⎠ 0
ˆ4 ) operates on an eigenvector, it permutes the first four and the last four When (reg) Γ(C elements among themselves, according to the permutation cycles (1432)(5768). This leads to two different eigencolumns ⎛
⎞ v1 ⎜λ v1 ⎟ ⎜ ⎟ ⎜λ2 v1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ λv1 ⎟ ⎜ ⎟, ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎝ 0 ⎠ 3
0
⎛
⎞ 0 ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ v5 ⎟ ⎜ 2 ⎟ ⎜ λ v5 ⎟ ⎜ ⎟ ⎝ λv5 ⎠ λ3 v5
for each value of λ = (1, i, −1, −i). Thus we obtain eight independent eigencolumns. We now choose the first type with v1 = 1 and for a particular λ. This gives a column with elements (1, λ3 , λ2 , λ, 0, 0, 0, 0). Now operate on this column by
(5)
P reg to obtain
(1 − λ2 )(1, −λ, −1, λ, 0, 0, 0, 0). This shows immediately that eigencolumns for λ = 1, −1 do not belong to the subspace (5) V . We choose one of the remaining eigenvalues, ζ = i, so that we obtain the column (1, −i, −1, i, 0, 0, 0, 0). ˆ2 ) and (reg) Γ(C ˆ43 ) we obtain a constant times the same When operating with (reg) Γ(C (reg) column. But operating with Γ(ˆ σ5 ) we obtain a new column (0, 0, 0, 0, 1, −1, −i, i)
Construction of the irreducible representations
178
which is independent of and even orthogonal to the first column. So ⎞ ⎛ 1 0 ⎜ −i 0⎟ ⎟ ⎜ ⎜−1 0⎟ ⎟ ⎜ ⎟ ⎜ i 0⎟ ⎜ (5) Φ = (1/2) ⎜ ⎟ ⎜ 0 1⎟ ⎟ ⎜ ⎜ 0 −1⎟ ⎟ ⎜ ⎝ 0 −i ⎠ 0 i Performing the projection of (7.13), we obtain the Irrep matrices: 1 0 i 0 −1 0 −i 0 , , , 0 1 0 −i 0 −1 0 i 0 1
1 , 0
0 −1
−1 , 0
0 i
−i , 0
0 −i
i . 0
Suppose that we had started with element C2 . The regular Rep of this element permutes the elements in an eigencolumn according to the cycles: (13) (24) (56) (78). This leads to four different types of columns with possible eigenvalues λ = 1, −1. After projection with (5) ˆ reg there remain the four vectors with λ = −1. So λ = −1 occurs twice in (5) Γ(C2 ) P and C2 is not a good element to start with.
Exercise 7.1 In Chapter 2 or 3 it was found that the matrix 1 1 1 √ S = 2 i −i ˆ32 . Use the inverse process to “undiagonalize” the set of six diagonalized the matrix C matrices found in Example 7.2 and show that this similarity transformation produces a set of six matrices that form an Irrep of the group C3v .
8 Product groups and product representations
8.1 Introduction So far, we have presented the concepts and methodology required for the construction of Irreps of finite groups. However, if we are to deal with groups of infinitely countable order, such as the space-groups, we find that new strategies are required for their construction. These strategies rely on the concepts of induction and subduction of group representations that we develop here and in the following chapter. First, we need to introduce new group theoretical entities such as subgroups, cosets, and invariant subgroups, upon which we construct the methodology for generating the Irreps of space-groups. In addition, as we will see, they take many guises in different applications.
8.2 Subgroups and cosets A subgroup H, of order h, of a group G, denoted by H ⊂ G, is a subset of h elements of G that is itself a group under the binary composition law associated with G. Two trivial subgroups of a given group G can be identified: G itself and the group of the identity E. Any other subgroup is called proper.
Example 8.1 The point-group C3v has four proper subgroups: C3 = {E, C3 , C32 }, and {E, σ1 }, {E, σ2 }, {E, σ3 } ≡ Cs .
We now use the concept of a subgroup to define cosets as follows: For H ⊂ G,
left coset right coset
Ri H, H Ri ,
179
* Ri ∈ G, Ri ∈ H.
(8.1)
180
Product groups and product representations
Based on this definition of left and right cosets, the following properties must hold for cosets: (i) The elements of H itself comprise the left (right) coset EH (HE). (ii) The left (right) coset Ri H (HRi ) contains the element Ri ; • its elements are all distinct from each other, for if two elements of the coset are equal: Ri Hk = Ri Hl
implies
=⇒
Hk = Hl ,
Hk , Hl ∈ H,
(8.2)
• its elements are different from those of H as implied by the definition of a group: Hk = Ri Hl =⇒ Ri = Hk Hl−1 ∈ H.
(8.3)
/ H, and Rj ∈ / Ri H, we find that the (iii) If we choose another element Rj ∈ G, Rj ∈ elements of the coset Rj H are also distinct from those of H and Ri H. Proof For Ri , Rj ∈ G, • We may have Ri−1 Rj = Hk ∈ H; then Rj = Ri Hk and Rj H = Ri Hk H = Ri H and the two left cosets are identical. • Conversely, if there is an element common between the two cosets, say Ri Hk = Rj Hl , then Ri−1 Rj = Hk Hl ∈ H. (iv) Every element of the group G is in some left (right) coset and no element is in more than one left (right) coset, so that any two cosets are either identical or disjoint. (v) Every left (right) coset contains the same number of elements, this number being equal to the order of H, h. (vi) Lagrange’s theorem If H is a subgroup of G, its order h is a factor of the order g of G. The integer g/h is called the index of H in G. If g is a prime number, then G has only trivial subgroups. Coset representatives Therefore, a group G can be decomposed into unique left (right) cosets of one of its subgroups H ⊂ G: Choose R1 = E, EH = H. Next, choose R2 from the set of elements G −H, followed by R3 from the set G − H − R2 H, and so on until all the elements of the group G are exhausted. We then obtain the decomposition
G =
g/h
Ri H.
(8.4)
i=1
The elements Ri are called coset representatives. A representative of coset i may be chosen as any element from coset i; however, once chosen it will be elevated to the level of representative and given the notation Ri .
8